We provide integrals practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on integrals skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.
List of integrals Questions
| Question No | Questions | Class |
|---|---|---|
| 1 | Evaluate: ( int_{0}^{pi} frac{x tan x}{sec x cdot operatorname{cosec} x} d x ) | 12 |
| 2 | ( lim _{n rightarrow infty} nleft[frac{1}{(n+1)(n+2)}+frac{1}{(n+2)(n+}right. ) is equal to ( ^{A} cdot log left(frac{3}{2}right) ) в. ( log left(frac{5}{2}right) ) c. ( log left(frac{1}{2}right) ) D. ( log left(frac{7}{4}right) ) | 12 |
| 3 | Illustration 2.42 Solve the integral I = GMm d. *2 | 12 |
| 4 | ( int e^{x}left(frac{1+sqrt{1-x^{2}} sin ^{-1} x}{sqrt{1-x^{2}}}right) d x= ) A ( cdot frac{e^{x}}{sqrt{1-x^{2}}}+c ) B . ( e^{x} sin ^{-1} x+c ) c. ( e^{x}left(e^{sin ^{-1} x}+frac{1}{sqrt{1-x^{2}}}right)+c ) D. ( e^{sin ^{-1} x}+frac{1}{sqrt{1-x^{2}}}+c ) | 12 |
| 5 | Write a value of ( int frac{sin x-cos x}{sqrt{1+sin 2 x}} d x ) | 12 |
| 6 | Solve: ( int frac{sin x-cos x}{sqrt{sin 2 x}} d x ) | 12 |
| 7 | Evaluate ( int frac{(sin x)^{2018}}{(cos x)^{2020}} d x ) A. ( frac{(tan x)^{2019}}{2019}+c ) B. ( frac{(sin x)^{2019}}{2019}+c ) c. ( frac{(cos x)^{2019}}{2019}+c ) D. ( frac{(tan x)^{2019} sec ^{2} x}{2019}+c ) | 12 |
| 8 | By Simpson’s rule, the value of ( int_{-3}^{3} x^{4} d y ) by taking 6 sub-intervals, is A . 98 B. 90 c. 80 D. 70 | 12 |
| 9 | Find: ( int frac{left(x^{4}-xright)^{4}}{x^{3}} d x ) | 12 |
| 10 | ( int frac{sqrt{1-x^{2}}+sqrt{1+x^{2}}}{sqrt{1-x^{4}}} d x= ) A ( cdot cosh ^{-1} x+sin ^{-1} x+c ) B. ( cosh ^{-1} x+cos ^{-1} x+c ) ( c cdot sinh ^{-1} x+sin ^{-1} x+c ) D. ( sinh ^{-1} x+cos ^{-1} x+c ) | 12 |
| 11 | ( int frac{x sin ^{-1} x}{sqrt{1-x^{2}}} d x= ) | 12 |
| 12 | 7. Evaluate: scos20″ dat Evaluate : sin x | 12 |
| 13 | 8. 52 sin x cos x dx is equal to (a) cos 2x + c (b) sin 2x + c (c) cos? x + c (d) sin? x + c | 12 |
| 14 | Resolve ( frac{6 x^{4}+11 x^{3}+18 x^{2}+14 x+6}{(x+1)left(x^{2}+x+1right)^{2}} ) into partial fractions. A ( frac{5}{x+1}+frac{(x-1)}{left(x^{2}+x+1right)}+frac{(3 x+2)}{left(x^{2}+x+1right)^{2}} ) в. ( frac{5}{x+1}-frac{(x-1)}{left(x^{2}+x+1right)}+frac{(3 x+2)}{left(x^{2}+x+1right)^{2}} ) c. ( frac{5}{x+1}+frac{(x-1)}{left(x^{2}+x+1right)}-frac{(3 x+2)}{left(x^{2}+x+1right)^{2}} ) D. ( frac{5}{x+1}+frac{(x+1)}{left(x^{2}+x+1right)}+frac{(3 x+2)}{left(x^{2}+x+1right)^{2}} ) | 12 |
| 15 | sin x 1. I dx = AX + B log sin(x-a),+C, then value of sin(x -a) (A,B) is [2004] (a) (-cos a, sin a) (b) (cos a, sina) (c) (-sin a, cos a) (d) (sin a, cos a) | 12 |
| 16 | ( int frac{boldsymbol{a}^{boldsymbol{x}}}{sqrt{mathbf{1}-boldsymbol{a}^{2 boldsymbol{x}}}} boldsymbol{d} boldsymbol{x}= ) A ( cdot frac{1}{log a} sin ^{-1}left(a^{x}right)+c ) B. ( frac{1}{log a} sinh left(a^{x}right)+c ) c. ( sin ^{-1}left(a^{x}right)+c ) D. ( log a sin ^{-1}left(a^{x}right)+c ) | 12 |
| 17 | ( int frac{boldsymbol{d} boldsymbol{x}}{(sqrt{mathbf{1}+boldsymbol{x}^{2}}-boldsymbol{x})^{n}}(boldsymbol{n} neq pm mathbf{1})= ) ( frac{1}{2}left(frac{z^{n+1}}{n+1}+frac{z^{n-1}}{n-1}right)+O ) where A. ( z=x-sqrt{1+x^{2}} ) B. ( z=sqrt{1+x^{2}}-x ) c. ( z=x+sqrt{1+x^{2}} ) D. ( z=x-sqrt{1-x^{2}} ) | 12 |
| 18 | Prove ( int_{0}^{a} boldsymbol{F} boldsymbol{d} boldsymbol{x}=int_{0}^{boldsymbol{a} / mathbf{2}} boldsymbol{F}(boldsymbol{x}) boldsymbol{d} boldsymbol{x}+ ) ( int_{0}^{boldsymbol{a} / 2} boldsymbol{F}(boldsymbol{a}-boldsymbol{x}) boldsymbol{d} boldsymbol{x} ) | 12 |
| 19 | Resolve into partial fraction ( frac{x^{3}-3 x-2}{left(x^{2}+x+1right)(x+1)^{2}} ) A ( cdot frac{3 x-1}{x^{2}-x+1}+frac{1}{(x+1)^{2}}-frac{3}{(x+1)} ) B. ( frac{3 x-1}{x^{2}-x+1}+frac{2}{(x+1)^{2}}-frac{3}{(x+1)} ) c. ( frac{3 x-1}{x^{2}+x+1}+frac{2}{(x+1)^{2}}-frac{3}{(x+1)} ) D. ( frac{3 x-1}{x^{2}+x+1}+frac{2}{(x+1)^{2}}+frac{3}{(x+1)} ) | 12 |
| 20 | The value of ( frac{mathbf{3 6}}{boldsymbol{pi}} int_{boldsymbol{pi} / boldsymbol{6}}^{boldsymbol{pi} / boldsymbol{3}} frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{1}+sqrt{cot boldsymbol{x}}} ) is | 12 |
| 21 | Solve:- ( sin ^{-1}(cos x) ) | 12 |
| 22 | Let ( a, b, c ) be non-zero real numbers such the : ( int_{0}^{1}left(1+cos ^{8} xright)left(a x^{2}+b x+cright) d x= ) ( int_{0}^{2}left(1+cos ^{8} xright)left(a x^{2}+b x+cright) d x, ) then the quadratic equation ( a x^{2}+b x+c= ) 0 has ( A cdot ) no root in (0,2) B. atleast one root in (0,2) C ( . ) a double root in (0,2) D. none | 12 |
| 23 | Integrate ( int_{2}^{3}left(2 x^{2}+1right) d x ) | 12 |
| 24 | ( int frac{1}{7} sin left(frac{x}{7}+10right) d x ) is equal to ( ^{mathrm{A}} cdot frac{1}{7}^{cos }left(frac{x}{7}+10right)+C ) B ( cdot-frac{1}{7} cos left(frac{x}{7}+10right)+C ) ( ^{mathbf{c}}-cos left(frac{x}{7}+10right)+C ) D ( -7 cos left(frac{x}{7}+10right)+C ) E ( cdot cos (x+70)+C ) | 12 |
| 25 | 10. 12 sin(x)dx is equal to: (a) -2 cos x + C (c) -2 cos x (b) 2 cos x + C (d) 2 cos x | 12 |
| 26 | ( int_{0}^{1} frac{log (1+x)}{1+x^{2}} d x= ) ( A cdot pi log 2 ) в. ( frac{pi}{8} log 2 ) c. ( frac{pi}{4} log 2 ) D. ( -pi log 2 ) | 12 |
| 27 | ( int frac{x^{3}}{sqrt{1+x^{2}}} d x ) A. ( quad sqrt{1+x}-frac{x}{3}left(1+x^{2}right)^{3 / 2}+c ) B ( cdot quad x sqrt{1+x^{2}}+frac{2}{3}left(1+x^{2}right)^{3 / 2}+c ) C ( cdot frac{x^{2} sqrt{1+x^{2}}}{3}-frac{2}{3} sqrt{1+x^{2}}+c ) D. ( quad x^{2} sqrt{1+x^{2}}-frac{1}{3}left(1+x^{2}right)^{3 / 2}+c ) | 12 |
| 28 | ( int_{log 2}^{t} frac{boldsymbol{d}_{boldsymbol{X}}}{sqrt{boldsymbol{e}^{boldsymbol{x}}-mathbf{1}}}=frac{boldsymbol{pi}}{boldsymbol{6}}, ) then ( mathbf{t}= ) ( A cdot 4 ) B. ( log 8 ) ( c cdot log 4 ) D. ( log 2 ) | 12 |
| 29 | Evaluate ( int_{0}^{pi / 4} frac{tan ^{3} x}{1+cos 2 x} d x ) | 12 |
| 30 | ( frac{boldsymbol{x}^{2}+mathbf{2} boldsymbol{x}+mathbf{3}}{boldsymbol{x}^{3}}=frac{boldsymbol{A}}{boldsymbol{x}}+frac{boldsymbol{B}}{boldsymbol{x}^{2}}+frac{boldsymbol{C}}{boldsymbol{x}^{3}} Rightarrow boldsymbol{A}+ ) ( boldsymbol{B}-boldsymbol{C}= ) ( A cdot 6 ) B. 3 ( c ) ( D ) | 12 |
| 31 | ( frac{2 x^{2}+2 x+1}{x^{3}+x^{2}}= ) A ( cdot frac{1}{x}-frac{1}{x^{2}}-frac{1}{x+1} ) в. ( frac{1}{x}-frac{1}{x^{2}}+frac{1}{x+1} ) c. ( frac{1}{x}+frac{1}{x^{2}}+frac{1}{x+1} ) D. ( frac{1}{x}+frac{1}{x^{2}}-frac{1}{x+1} ) | 12 |
| 32 | ( int frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{x}left(boldsymbol{x}^{5}+boldsymbol{3}right)} ) | 12 |
| 33 | If ( I=int frac{(x+1)^{2}}{sqrt{x^{2}+1}} d x, ) then 2 l equals A ( cdot(x+4) sqrt{x^{2}+1}+log (x+sqrt{x^{2}+1})+C ) B. ( x sqrt{x^{2}+1}+2 log (x+sqrt{x^{2}+1})+C ) c. ( x sqrt{x^{2}+1}+log (x+sqrt{x^{2}+1})+C ) D. ( (x-3) sqrt{x^{2}+1}+log (x+sqrt{x^{2}+1})+C ) | 12 |
| 34 | ( int frac{x}{1+cos x} d x= ) ( mathbf{A} cdot x tan frac{x}{2}-2 log |sec x / 2|+c ) B. ( -x tan x / 2-frac{1}{2} log |sec x / 2|+c ) c. ( _{x tan x / 2+frac{1}{2} log |sec x / 2|+c} ) D. ( x cot x / 2-frac{1}{2} log |csc x / 2|+c ) | 12 |
| 35 | ( int x sec ^{2} x d x ) | 12 |
| 36 | Solve : ( boldsymbol{I}=int sin ^{6} boldsymbol{x} boldsymbol{d} boldsymbol{x} ) | 12 |
| 37 | If ( int frac{1}{1+cot x} d x=A log mid sin x+cos ) ( boldsymbol{x} mid+mathrm{Bx}+c, ) then ( boldsymbol{A}=ldots ldots . ., boldsymbol{B}= ) A ( cdot-frac{1}{2}, frac{1}{2} ) в. -1,1 ( c cdot frac{1}{3}, frac{1}{2} ) D. ( frac{-1}{3}, frac{1}{2} ) | 12 |
| 38 | ( int frac{boldsymbol{d} boldsymbol{x}}{mathbf{1}+mathbf{3} boldsymbol{e}^{boldsymbol{x}}+boldsymbol{2} boldsymbol{e}^{boldsymbol{2} boldsymbol{x}}} cdot ) A ( cdot log frac{e^{x}left(1-e^{x}right)}{left(1+2 e^{x}right)^{2}} ) B ( cdot log frac{e^{x}left(1+e^{x}right)}{left(1+e^{x}right)^{2}} ) C ( cdot log frac{e^{x}left(1+e^{x}right)}{left(1+2 e^{x}right)^{2}} ) D. ( log frac{e^{x}left(1+e^{x}right)}{left(1-2 e^{x}right)^{2}} ) | 12 |
| 39 | Evaluate ( int_{0}^{pi / 2} frac{cos x}{left(1+sin ^{2} xright)} d x ) A ( cdot frac{pi}{2} ) B. ( frac{pi}{4} ) c. ( pi ) D. None of these | 12 |
| 40 | ( int_{1}^{2} frac{d x}{left(x^{2}-2 x+4right)^{frac{3}{2}}}=frac{k}{k+5}, ) then ( k ) is equal to A . 1 B. 2 ( c .3 ) D. 4 | 12 |
| 41 | Evaluate the following integrals: ( int frac{1}{sqrt{7-3 x-2 x^{2}}} d x ) | 12 |
| 42 | Evaluate the integral ( int_{0}^{1} x^{2} e^{x} d x ) ( mathbf{A} cdot e-2 ) B. ( e+2 ) ( c ) D. ( e+3 ) | 12 |
| 43 | ( int_{pi^{2} / 16}^{pi^{2} / 4} frac{sin sqrt{x}}{sqrt{x}} d x= ) A ( cdot sqrt{2} ) B. ( 1 / sqrt{2} ) ( c cdot 2 sqrt{2} ) D. ( pi / 2 ) | 12 |
| 44 | ( I=int e^{x} frac{(2+sin 2 x)}{(1+cos 2 x)} d x ) A ( cdot e^{x} sin x ) B. ( e^{x} cos x ) ( mathbf{c} cdot e^{x} tan x ) D. ( e^{x} cos 2 x ) | 12 |
| 45 | ( int sqrt{left(frac{x-1}{x+1}right)} d x ) | 12 |
| 46 | Evaluate: ( int frac{(x(pi+49))^{15 / 7}}{pi^{2}left(x^{pi}+7right)} d x ) | 12 |
| 47 | Integrate with respect to ( x ) ( frac{1-sin x}{x+cos x} ) | 12 |
| 48 | ( underset{boldsymbol{n} rightarrow infty}{boldsymbol{L} boldsymbol{t}} sum_{boldsymbol{r}=0}^{boldsymbol{n}-mathbf{1}} frac{boldsymbol{n}}{boldsymbol{n}^{2}+boldsymbol{r}^{2}} ) ( mathbf{A} cdot mathbf{1} ) B. ( c cdot frac{pi}{2} ) D. ( frac{pi}{4} ) | 12 |
| 49 | For any integer ( n ) the integral ( int_{0}^{pi} e^{cos ^{2} x} cos ^{3}(2 n+1) x d x ) has the value A . ( pi ) B. c. 0 D. none of these | 12 |
| 50 | ( int_{1}^{e^{37}} frac{pi sin left(pi log _{e} xright)}{x} d x ) is equal to A . 2 B. – c. ( 2 / pi ) D. ( 2 pi ) | 12 |
| 51 | Evaluate: ( int_{1}^{2} log x d x ) A. ( 2 log 2-1 ) B. ( log 2-1 ) c. ( 2 log 2+1 ) D. ( log 2-2 ) | 12 |
| 52 | Evaluate ( int frac{1}{a^{x} b^{x}} d x ) | 12 |
| 53 | The value of ( int_{0}^{pi / 4} log (1+tan x) d x ) is equal to A ( cdot frac{pi}{8} log _{e} 2 ) в. ( frac{pi}{4} log _{e} 2 ) c. D. none of these | 12 |
| 54 | Find the integral of the function ( frac{cos x-sin x}{1+sin 2 x} ) | 12 |
| 55 | The value of the definite integral, ( int_{0}^{pi / 2} frac{sin 5 x}{sin x} d x ) is A . в. ( c . pi ) D . ( 2 pi ) | 12 |
| 56 | Evaluate ( int_{0}^{x}[cos t] d t, ) where ( n in ) ( left(2 n pi,(4 n+1) frac{pi}{2}right), n in N, ) and denotes the greatest integer function. | 12 |
| 57 | Find the interval in which ( boldsymbol{f}(boldsymbol{x})=int_{-1}^{x}left(boldsymbol{e}^{t}-mathbf{1}right)(mathbf{2}-boldsymbol{t}) boldsymbol{d t},(boldsymbol{x}>mathbf{1}) ) is increasing ( A cdot[3,5] ) B . [1,3] ( mathbf{c} cdot[0,3] ) D. [0,2] | 12 |
| 58 | The value of ( int_{0}^{pi / 2} frac{f(x) d x}{f(x)+f(pi / 2-x)} ) is equal to A . ( pi / 4 ) B . ( pi / 2 ) ( c . pi ) D. None | 12 |
| 59 | Evaluate: ( int frac{x^{2}}{left(x^{2}+2right)left(2 x^{2}+1right)} d x ) | 12 |
| 60 | ( int_{0}^{1} sqrt{boldsymbol{x}(mathbf{1}-boldsymbol{x})} boldsymbol{d} boldsymbol{x} ) | 12 |
| 61 | The value of the definite integral ( int frac{d theta}{1+tan theta}=frac{501 pi}{K} ) where ( a_{2}=frac{1003 pi}{2008} ) and ( boldsymbol{a}_{1}=frac{pi}{2008} ) The value of ( mathrm{K} ) equalls A. 2007 B. 2006 c. 2009 D. 2008 | 12 |
| 62 | Let ( f(x)=sqrt{3 x-3} ) and ( c ) be the number that satisfies the Mean value theorem for ( f ) on the interval [4,13] What is the value of ( c ) ? A . 11.5 в. 7.75 c. 7.5 D. 5.5 | 12 |
| 63 | ( int(x+2) sqrt{x^{2}+1} d x ) | 12 |
| 64 | The value of ( intleft(x e^{ell n sin x}-cos xright) d x ) is equal to: ( mathbf{A} cdot x cos x+C ) B. ( sin x-x cos +C ) c. ( -e^{e n x} cos x+C ) ( mathbf{D} cdot sin x+x cos x+C ) | 12 |
| 65 | Evaluate ( int e^{x} sin e^{x} d x ) on ( R ) | 12 |
| 66 | Resolve ( frac{x}{(1+x)left(1+x^{2}right)^{2}} ) into partial fractions. A ( cdot frac{-1}{4(1+x)}+frac{(x-1)}{4left(1+x^{2}right)}+frac{(x+1)}{2left(1+x^{2}right)^{2}} ) B. ( frac{1}{4(1+x)}+frac{(x-1)}{4left(1+x^{2}right)}+frac{(x+1)}{left(1+x^{2}right)^{2}} ) C ( frac{1}{2(1+x)}+frac{(x-1)}{2left(1+x^{2}right)}+frac{(x+1)}{2left(1+x^{2}right)^{2}} ) D ( frac{1}{4(1+x)}-frac{(x-1)}{4left(1+x^{2}right)}+frac{(x+1)}{2left(1+x^{2}right)^{2}} ) | 12 |
| 67 | The integral ( int frac{d x}{a cos x+b sin x} ) is of the form ( frac{1}{r} ln left[tan left(frac{x+alpha}{2}right)right] ) What is ( alpha ) equal to? A ( cdot tan ^{-1}left(frac{a}{b}right) ) B. ( tan ^{-1}left(frac{b}{a}right) ) c. ( tan ^{-1}left(frac{a+b}{a-b}right) ) D. ( tan ^{-1}left(frac{a-b}{a+b}right) ) | 12 |
| 68 | Evaluate the definite integral: ( int_{0}^{pi / 2} sin x d x ) | 12 |
| 69 | Integrate : ( intleft(x^{4}-x^{2}+1-frac{2}{1+x^{2}}right) d x ) | 12 |
| 70 | Find the integral of ( intleft(2 x^{2}-3 sin x+right. ) ( mathbf{5} sqrt{boldsymbol{x}}) boldsymbol{d} boldsymbol{x} ) | 12 |
| 71 | If ( f(x)=int frac{left(x^{2}+sin ^{2} xright)}{1+x^{2}} sec ^{2} x d x ) and ( f(0)=0 ) then ( f(1) ) is equal to A ( cdot 1-frac{pi}{4} ) B. ( frac{pi}{4}-1 ) c. ( tan 1-frac{pi}{4} ) D. ( frac{pi}{4}-tan 1 ) | 12 |
| 72 | Resolve into partial fractions ( frac{x^{2}+2}{(x+1)^{3}(x-2)} ) A ( cdot frac{6}{(x+2)}+frac{6}{(x+1)}-frac{5}{(x+1)^{2}}+frac{3}{(x+1)^{3}} ) B. ( -frac{6}{(x+2)}+frac{6}{(x+1)}-frac{5}{(x+1)^{2}}+frac{3}{(x+1)^{3}} ) C ( cdot-frac{6}{(x+2)}+frac{3}{(x+1)}-frac{5}{(x+1)^{2}}+frac{3}{(x+1)^{3}} ) D ( cdot-frac{3}{(x+2)}-frac{3}{(x+1)}-frac{5}{(x+1)^{2}}+frac{3}{(x+1)^{3}} ) | 12 |
| 73 | Evaluate : ( int_{0}^{pi} frac{x tan x}{sec x+tan x} ) | 12 |
| 74 | Evaluate the integral ( int_{0}^{1 / 2} frac{x sin ^{-1} x}{sqrt{1-x^{2}}} d x ) | 12 |
| 75 | Evaluate: ( intleft(sqrt{boldsymbol{x}}+frac{mathbf{1}}{sqrt{boldsymbol{x}}}right) boldsymbol{d} boldsymbol{x} ) | 12 |
| 76 | Solve: ( int frac{1+x+sqrt{x+x^{2}}}{sqrt{x}+sqrt{1+x}} d x ) is equal to A ( cdot frac{1}{2} sqrt{1+x} C ) B ( cdot frac{2}{3}(1+x)^{3 / 2}+C ) ( mathbf{c} cdot sqrt{1+x}+C ) D ( cdot frac{3}{2}(1+x)^{32}+C ) | 12 |
| 77 | Evaluate: ( int_{0}^{frac{pi}{2}} frac{sin x}{sin x+cos x} d x ) | 12 |
| 78 | ( int e^{tan ^{-1} x}left[frac{1+x+x^{2}}{1+x^{2}}right] d x= ) A ( cdot x^{2} e^{tan ^{-1} x}+c ) B . ( x e^{tan ^{-1} x}+c ) C ( cdot e^{tan ^{-1} x}+c ) D. ( frac{1}{2} e^{tan ^{-1} x}+c ) | 12 |
| 79 | Evaluate: ( int frac{sin x-cos x}{sqrt{sin 2 x}} d x ) | 12 |
| 80 | Show that ( int sqrt{4+8 x-5 x^{2}} d x= ) ( sqrt{5}left[frac{5 x-4}{10 sqrt{(5)}} sqrt{4+8 x-5 x^{2}}+frac{18}{25} sin ^{-1} 1right. ) | 12 |
| 81 | Evaluate the given integral. ( int e^{x}left(frac{1+x}{(2+x)^{2}}right) d x ) | 12 |
| 82 | The value of the integral ( int_{-pi / 4}^{pi / 4} log (sec theta-tan theta) d theta ) is ( A cdot frac{pi}{4} ) B. ( c cdot 0 ) ( D ) | 12 |
| 83 | ( int 3^{x} cos 5 x d x= ) A. ( frac{3^{x}}{(log 3)^{2}+25}[(log 3) cdot cos 5 x-5 sin 5 x]+c ) B. ( frac{3^{x}}{(log 3)^{2}+25}[(log 3) cdot cos 5 x+5 sin 5 x]+c ) C. ( frac{3^{x}}{(log 3)^{2}+25}[5 cos 5 x-(log 3) cdot sin 5 x]+c ) D. ( frac{3^{x}}{(log 3)^{2}+25}[5 cos 5 x+(log 3) cdot sin 5 x]+c ) | 12 |
| 84 | If ( int frac{sec x-tan x}{sqrt{sin ^{2} x-sin x}} d x=k ln mid f(x)+ ) ( sqrt{2} sqrt{tan x(tan x-sec x)} mid+c, ) where ( c ) is arbitrary constant and ( k ) is a fixed constant, then This question has multiple correct options A. ( k=sqrt{2} ) в. ( k=frac{1}{sqrt{2}} ) c. ( f(x)=tan x-sec x ) D. ( f(x)=sqrt{tan x+sec x} ) | 12 |
| 85 | Evaluate the integral ( int_{0}^{1}left(3 x^{2}+2 xright) d x ) | 12 |
| 86 | ( int sec ^{2} x log left(1+sin ^{2} xright) d x= ) ( tan x log left(1+sin ^{2} xright)-2 x+ ) ( sqrt{k} tan ^{-1} sqrt{k} tan x . ) Find the value of ( k ) | 12 |
| 87 | Solve ( : int frac{2 x^{3}-3 x^{2}-8 x-26}{2 x^{2}-5 x+2} d x ) | 12 |
| 88 | ( int e^{e^{x}+x} d x= ) A ( cdot e^{e^{x}}+x+c ) B . ( e^{e^{x}}+c ) ( mathbf{c} cdot e^{x}+c ) ( mathbf{D} cdot e^{x}+x+c ) | 12 |
| 89 | Find ( int frac{d x}{sqrt{9+8 x-x^{2}}} ) | 12 |
| 90 | ( frac{x^{2}}{left(x^{2}+1right)left(x^{2}+4right)}=frac{A}{x^{2}+1}+ ) ( frac{B}{x^{2}+4} Rightarrow A+B= ) A. B. ( c cdot 2 ) D. 3 | 12 |
| 91 | 26. Let F(x)=f(r)+S) where f(x) = f log de, Then Fle) equals [2007] (a) 1 (b) 2 (c) 1/2 (d) | 12 |
| 92 | Evaluate the given integral: ( int_{0}^{1}(1+ ) ( x)^{5} d x ) | 12 |
| 93 | Evaluate : ( int_{0}^{pi / 2} x cos 2 x d x ) | 12 |
| 94 | ( lim _{n rightarrow infty} frac{sqrt{mathbf{1}}+sqrt{mathbf{2}}+ldots ldots+sqrt{n-1}}{boldsymbol{n} sqrt{boldsymbol{n}}}=mathbf{0} ) A ( cdot frac{1}{2} ) B. ( c cdot frac{1}{3} ) D. 0 (zero) | 12 |
| 95 | If ( int x^{5} e^{-x^{2}} d x=g(x) cdot e^{-x^{2}}+C ) then the value of ( g(-1) ) is? ( A cdot frac{3}{2} ) в. ( frac{5}{2} ) ( c cdot-frac{5}{2} ) D. | 12 |
| 96 | Solve ( : int_{0}^{1} e^{e^{x}}left(1+x cdot e^{x}right) d x ) | 12 |
| 97 | Resolve into partial fractions ( boldsymbol{x}^{mathbf{3}} ) ( (x-1)^{4}left(x^{2}-x+1right) ) A ( cdot frac{1}{(x-1)^{4}}+frac{2}{(x-1)^{2}}+frac{x}{x^{2}-x+1} ) B. ( frac{1}{(x-1)^{4}}-frac{1}{(x-1)^{2}}-frac{1}{(x-1)}+frac{x}{x^{2}-x+1} ) C ( frac{1}{(x-1)^{4}}+frac{2}{(x-1)^{2}}-frac{1}{(x-1)}+frac{x}{x^{2}-x+1} ) D ( frac{-1}{(x-1)^{4}}+frac{2}{(x-1)^{2}}-frac{1}{(x-1)}+frac{x}{x^{2}+x+1} ) | 12 |
| 98 | ( int_{0}^{pi / 2} sin 2 x tan ^{-1}(sin x) d x= ) A. ( frac{pi}{2}-1 ) B. ( frac{pi}{2}+1 ) c. ( frac{3 pi}{2}+1 ) D. ( frac{3 pi}{2}-1 ) | 12 |
| 99 | If linear function ( f(x) ) and ( g(x) ) satisfy ( int[(3 x-1) cos x+(1-2 x) sin x] d x= ) ( boldsymbol{f}(boldsymbol{x}) cos boldsymbol{x}+boldsymbol{g}(boldsymbol{x}) sin boldsymbol{x}+boldsymbol{C}, ) then A. ( f(x)=3 x-3 ) B. ( g(x)=3+x ) C. ( f(x)=3(x-1) ) D. ( g(x)=3(x-1) ) | 12 |
| 100 | Evaluate : ( int frac{1}{sin ^{2} x cos ^{2} x} d x ) A . ( -tan 2 x+C ) B. ( -2 cot 2 x+C ) ( c cdot tan x+cot 2 x+C ) D. None of these | 12 |
| 101 | ( int e^{x}left[f(x)+f^{prime}(x)right] d x ) is equal to ( mathbf{A} cdot e^{x} f(x)+c ) B ( cdot e^{x}+c ) ( mathbf{c} cdot e^{x} f^{prime}(x)+c ) D. None of these | 12 |
| 102 | ( int frac{ln (1+x)}{1+x} d x ) equals A ( cdot frac{(ln (1+x))^{2}}{2} ) B . ( -pi ln (1+x) ) c. ( frac{pi}{2} ln (1+x) ) D. ( -frac{pi}{2} ln (1+x) ) | 12 |
| 103 | Solve: ( int_{frac{pi}{6}}^{frac{pi}{3}} frac{sin x+cos x}{sqrt{sin 2 x}} d x ) | 12 |
| 104 | Solve : ( int e^{x} cdot sin 3 x d x ) | 12 |
| 105 | If ( boldsymbol{I}=int_{0}^{pi} x^{3} log sin x d x ) and ( I= ) ( int_{0}^{pi} x^{2} log (sqrt{2} sin x), ) then the value of ( frac{4}{3 pi} I ) is equal to | 12 |
| 106 | Evaluate the following integral: ( int_{0}^{pi} x d x ) | 12 |
| 107 | If ( fleft(frac{3 x-4}{3 x+4}right)=x+2, x neq-frac{4}{3}, ) and ( int boldsymbol{f}(boldsymbol{x}) boldsymbol{d} boldsymbol{x}=boldsymbol{A} log |mathbf{1}-boldsymbol{x}|+boldsymbol{B} boldsymbol{x}+boldsymbol{C} ) then the ordered pair ( (A, B) ) is equal to (where ( C ) is a constant of integration) ( ^{A} cdotleft(frac{8}{3}, frac{2}{3}right) ) B ( cdotleft(-frac{8}{3}, frac{2}{3}right) ) ( ^{mathbf{C}} cdotleft(-frac{8}{3},-frac{2}{3}right) ) D. ( left(frac{8}{3},-frac{2}{3}right) ) | 12 |
| 108 | Integrate the rational function ( frac{3 x+5}{x^{3}-x^{2}-x+1} ) | 12 |
| 109 | Evaluate: ( int sqrt{tan x} d x,left(0<x<frac{pi}{2}right) ) | 12 |
| 110 | Evaluate : ( int_{frac{pi}{6}}^{frac{pi}{3}} frac{boldsymbol{d x}}{1+sqrt{tan x}} ) A ( cdot frac{pi}{4} ) в. c. ( frac{pi}{12} ) D. | 12 |
| 111 | The value of ( int frac{cos sqrt{x}}{sqrt{x}} d x ) is ( A cdot 2 cos sqrt{x}+C ) B. ( sqrt{frac{cos x}{x}}+C ) ( c cdot sin sqrt{x}+C ) D. ( 2 sin sqrt{x}+C ) | 12 |
| 112 | ( cos boldsymbol{x} cdot log (cos boldsymbol{x}) boldsymbol{d} boldsymbol{x}= ) A ( . sin x log (cos x)-log (cos x)+c ) B. ( sin x log (cos x)+sec x+c ) c. ( sin x log (cos x)-sin x+log |sec x+tan x|+c ) D. ( sin x log (cos x)-sec x+c ) | 12 |
| 113 | Integrate the rational function ( frac{1}{x^{4}-1} ) | 12 |
| 114 | ( int_{0}^{pi / 2} frac{sin x}{sqrt{1+cos x}} d x= ) A ( cdot sqrt{2}-1 ) B. ( 2 sqrt{2} ) c. ( 2(sqrt{2}-1) ) D. ( frac{sqrt{2}+1}{2} ) | 12 |
| 115 | The solution of the equation ( frac{d y}{d x}= ) ( frac{x(2 log x+1)}{sin y+y cos y} ) is A ( y sin y=x^{2} log x+frac{x^{2}}{y}+c ) B ( cdot y cos y=x^{2}(log x+1)+c ) c. ( y cos y=x^{2} log x+frac{x^{2}}{2}+c ) D. ( y sin y=x^{2} log x+c ) | 12 |
| 116 | If ( I=int_{0}^{2 pi} e^{x / 2} sin left(frac{x}{2}+frac{pi}{4}right) d x, ) then ( I ) equals ( A ) B. ( mathbf{c} cdot-pi / 2 ) D. ( 2 pi ) | 12 |
| 117 | Evaluate ( int_{0}^{infty} frac{x^{2}+1}{x^{4}+7 x^{2}+1} d x ) ( A ) в. ( frac{pi}{2} ) c. D. | 12 |
| 118 | ( int_{0}^{pi / 2} frac{1}{1+sqrt{tan x}} d x ) A . 0 в. ( c cdot frac{pi}{4} ) D. ( -frac{pi}{4} ) | 12 |
| 119 | ( int^{-1}(f(x))=x ) | 12 |
| 120 | ( int frac{cos ^{2} x}{sin ^{4} x} d x ) A. ( -frac{1}{3} tan ^{3} x ) B. ( frac{1}{3} cot ^{3} x ) C. ( -frac{1}{3} cot ^{3} x ) D. ( frac{1}{3} tan ^{3} x ) | 12 |
| 121 | ( int_{0}^{1} x(1-x)^{4} d x= ) A . ( 1 / 15 ) B. 1/30 ( c cdot-1 / 15 ) D. 1/60 | 12 |
| 122 | Integrate the rational function ( frac{x}{(x-1)(x-2)(x-3)} ) | 12 |
| 123 | ( int frac{d t}{(6 t-1)} ) is equal to: A ( cdot frac{1}{6} ln (6 t-1)+C ) B. ( ln (6 t-1)+C ) ( c cdot-frac{1}{6} ln (6 t-1)+C ) D. None of these | 12 |
| 124 | 12. Integrateſ xº+3x+2_dx. + 2 | 12 |
| 125 | If differential equation of family of curves ( boldsymbol{y} ln |boldsymbol{c} boldsymbol{x}|=boldsymbol{x}, ) where ( c ) is an arbitrary constant, is ( boldsymbol{y}^{prime}=frac{boldsymbol{y}}{boldsymbol{x}}+boldsymbol{phi}left(frac{boldsymbol{x}}{boldsymbol{y}}right) ) for some function ( phi ), then ( phi(2) ) is equal to? | 12 |
| 126 | ( int_{0}^{pi / 2} frac{1}{1+sqrt[4]{tan x}} d x= ) A . ( pi / 4 ) в. ( pi / 3 ) ( c cdot 0 ) D. None of these | 12 |
| 127 | Suppose ( J=int frac{sin ^{2} x+sin x}{1+sin x+cos x} d x ) and ( K=int frac{cos ^{2} x+cos x}{1+sin x+cos x} d x . ) If ( C ) is an arbitrary constant of integration then which of the following is correct? A ( cdot J=frac{1}{2}(x-sin x+cos x)+C ) B. ( J=K-(sin x+cos x)+C ) ( mathbf{c} . J=x+K+C ) D. ( K=frac{1}{2}(x-sin x+cos x)+C ) | 12 |
| 128 | Evaluate the following: ( int(3 x+1) sqrt{2 x-1} d x ) | 12 |
| 129 | Assertion Consider the function ( boldsymbol{F}(boldsymbol{x})= ) ( int frac{x}{(x-1)left(x^{2}+1right)} d x ) STATEMENT-1 : ( boldsymbol{F}(boldsymbol{x}) ) is discontinuous at ( boldsymbol{x}=mathbf{1} ) Reason STATEMENT-2 : Integrand of ( boldsymbol{F}(boldsymbol{x}) ) is discontinuous at ( x=1 ) A. STATEMENT-1 is True, STATEMENT-2 is True: STATEMENT-2 is a correct explanation for STATEMENT- B. STATEMENT-1 is True, STATEMENT-2 is True: STATEMENT-2 is NOT a correct explanation for STATEMENT-1 c. STATEMENT-1 is True, STATEMENT-2 is False D. STATEMENT-1 is False, STATEMENT-2 is True | 12 |
| 130 | Integrate: ( frac{mathbf{3} boldsymbol{x}-mathbf{1}}{(boldsymbol{x}+mathbf{2})^{2}} ) | 12 |
| 131 | ( int frac{boldsymbol{d x}}{left(x^{2}+1right)left(x^{2}+4right)}= ) A ( cdot frac{1}{3} tan ^{-1} x-frac{1}{3} tan ^{-1} frac{x}{2}+c ) B – ( frac{1}{3} tan ^{-1} x+frac{1}{3} tan ^{-1} frac{x}{2}+c ) c. ( frac{1}{3} tan ^{-1} x-frac{1}{6} tan ^{-1} frac{x}{2}+c ) ( mathbf{D} cdot tan ^{-1} x-2 tan ^{-1} frac{x}{2}+c ) | 12 |
| 132 | if ( int f(x) d x=f(x), ) then ( intleft(frac{f(x)}{f^{prime}(x)}right) . d x ) is equal to A. ( x+c ) B. ( log f(x)+c ) ( c cdot log F(x)+c ) D. ( e^{f(x)}+c ) | 12 |
| 133 | Number of partial fractions obtained ( frac{3 x-5}{(x+1)^{3}left(x^{2}+1right)^{2}} ) A. 5 B. 4 ( c cdot 6 ) D. 3 | 12 |
| 134 | Solve: ( int frac{boldsymbol{d x}}{boldsymbol{x}left(boldsymbol{a}+boldsymbol{b} boldsymbol{x}^{n}right)^{2}} ) | 12 |
| 135 | Evaluate the integral ( int frac{2 x+3}{sqrt{x^{2}+4 x+1}} d x ) A ( cdot 2 sqrt{x^{2}+4 x+1}-log |x+2+sqrt{x^{2}+4 x+1}|+C ) B. ( sqrt{x^{2}+4 x-1}-log |x+2+sqrt{x^{2}+4 x-1}|+C ) c. ( 2 sqrt{x^{2}+4 x+1}-log |x-2+sqrt{x^{2}-4 x+1}|+C ) D. ( sqrt{x^{2}+4 x-1}-log |x-2+sqrt{x^{2}+4 x-1}|+C ) | 12 |
| 136 | Derive partial fraction for ( frac{5 x^{2}+1}{x^{3}-1}= ) A ( cdot frac{3}{x-1}+frac{2 x+1}{x^{2}+x+1} ) в. ( frac{4}{x-1}+frac{5 x+1}{x^{2}+x+1} ) c. ( frac{2}{x-1}+frac{3 x+1}{x^{2}+x+1} ) D. ( frac{1}{x-1}+frac{4 x+1}{x^{2}+x+1} ) | 12 |
| 137 | ( intleft(x^{2}-x+5right) d x ) A. ( frac{x^{3}}{3}-frac{x^{2}}{2}+5 x+c ) B. ( frac{x^{3}}{3}+frac{x^{2}}{2}+5 x+c ) c. ( frac{x^{2}}{2}-frac{x}{2}+5 x+c ) D. ( frac{x^{4}}{4}-frac{x^{4}}{3}+5 ) | 12 |
| 138 | If ( I=int_{1 / pi}^{pi} frac{1}{x} cdot sin left(x-frac{1}{x}right) d x, ) then ( I ) is equal to A . 0 в. ( pi ) c. ( _{pi-frac{1}{pi}} ) D. ( pi+frac{1}{pi} ) | 12 |
| 139 | ( int frac{1}{1-cos frac{x}{2}} d x ) | 12 |
| 140 | ( int_{0}^{frac{pi}{4}} frac{sin ^{2} x cdot cos ^{2} x}{left(sin ^{3} x+cos ^{3} xright)^{2}} d x=frac{m}{6} cdot ) Find ( m ) | 12 |
| 141 | ( int frac{left(t^{2}+1right)^{2}}{t^{6}+1} d t ) | 12 |
| 142 | ( int_{0}^{1} sqrt{boldsymbol{x}(1-boldsymbol{x})} boldsymbol{d} boldsymbol{x}= ) ( mathbf{A} cdot pi / 2 ) в. ( pi / 4 ) c. ( pi / 6 ) D . ( pi ) | 12 |
| 143 | If ( I=int_{0}^{pi} frac{x^{2} sin ^{2} x cos ^{4} x}{x^{2}-3 pi x+3 x^{2}} d x ) then the value of ( frac{32}{pi^{2}} I+298 ) is equal | 12 |
| 144 | Evaluate: ( int frac{cos 2 x}{sin x} d x ) | 12 |
| 145 | dx 4. Jos x+ 53 sin x equals cos x + 13 sin (a) log tan (3 + .) +C (6) log tan (.)+c cas) los tan ( 0)+c | 12 |
| 146 | ( int_{0}^{frac{pi}{2}} frac{sin x}{1+cos ^{2} x} d x ) | 12 |
| 147 | Obtain ( int_{2}^{3}(3 x+8) d x ) as limit of sum. | 12 |
| 148 | ( int_{0}^{pi}|cos x| d x=? ) ( A cdot 2 ) B. ( frac{3}{2} ) c. 1 D. | 12 |
| 149 | Evaluate ( int_{0}^{1} frac{mathbf{1}-boldsymbol{x}}{mathbf{1}+boldsymbol{x}} cdot frac{boldsymbol{d} boldsymbol{x}}{sqrt{boldsymbol{x}+boldsymbol{x}^{2}+boldsymbol{x}^{3}}} ) A ( cdot frac{pi}{3} ) в. c. ( frac{pi}{12} ) D. | 12 |
| 150 | ( int sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x=f(x)-log (1+ ) ( left.boldsymbol{x}^{2}right)+boldsymbol{c} ) then ( boldsymbol{f}(boldsymbol{x})= ) A ( cdot 2 x tan ^{-1} x ) B. ( -2 x tan ^{-1} x ) c. ( x tan ^{-1} x ) D. ( -x tan ^{-1} x ) | 12 |
| 151 | ( int x^{2} e^{x} d x=? ) | 12 |
| 152 | Evaluate ( int frac{d x}{left(a^{2}+x^{2}right)^{3 / 2}} ) ( boldsymbol{I}=frac{boldsymbol{x}}{boldsymbol{a}^{2}left(boldsymbol{x}^{2}+boldsymbol{a}^{2}right)^{boldsymbol{K}}}+boldsymbol{c} ) What is K? | 12 |
| 153 | Let ( f ) be a function defined for every ( x ) such that ( f^{prime prime}=-f, f(0)=0, f^{prime}(0)=1, ) then ( f(x) ) is equal to A. ( tan x ) B ( cdot e^{x}-1 ) ( c cdot sin x ) ( D cdot 2 sin x ) | 12 |
| 154 | 3. | (x ² ax is (a) 2-2 (c) V2 – 1 (6) 2+ √2 (d) -√2-√3+5 | 12 |
| 155 | Find ( boldsymbol{F}(boldsymbol{x}) ) from the ( operatorname{given} boldsymbol{F}^{prime}(boldsymbol{x}) ) ( boldsymbol{F}^{prime}(boldsymbol{x})=mathbf{4} boldsymbol{x}+mathbf{1} ) and ( boldsymbol{F}(-1)=mathbf{2} ) | 12 |
| 156 | ( boldsymbol{l} boldsymbol{t}_{n rightarrow infty}left[frac{mathbf{1}}{boldsymbol{n a}}+frac{mathbf{1}}{boldsymbol{n a}+mathbf{1}} frac{mathbf{1}}{boldsymbol{n a}+mathbf{2}} cdots cdots+frac{mathbf{1}}{boldsymbol{n b}}right] ) ( ^{mathrm{A}} cdot log left(frac{b}{a}right) ) B. ( log left(frac{a}{b}right) ) ( c cdot log a ) D. ( log b ) | 12 |
| 157 | Evaluate : ( boldsymbol{I}=int frac{boldsymbol{2} boldsymbol{x}}{boldsymbol{x}^{2}-boldsymbol{6} boldsymbol{x}+boldsymbol{6}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 158 | If ( int frac{2 x^{2}+3}{left(x^{2}-1right)left(x^{2}+4right)} d x= ) ( a log left(frac{x-1}{x+1}right)+b tan ^{-1}left(frac{x}{2}right)+c, ) then values of a and ( b ) are A. (1,-1) в. (-1,1) c. ( left(frac{1}{2},-frac{1}{2}right. ) D. ( left(frac{1}{2}, frac{1}{2}right) ) | 12 |
| 159 | Evaluate ( int(ln x+1) d x ) ( mathbf{A} cdot x ln x+c ) B. ( x^{2} ln x+c ) c. ( x^{-2} ln x+c ) D. ( -x ln x+c ) | 12 |
| 160 | If ( I_{n}=int_{0}^{pi / 4} tan ^{n} d x, ) then ( frac{1}{I_{2}+I_{4}} frac{1}{I_{3}+I_{5}} frac{1}{I_{4}+I_{6}} ) is: A. A.P B. G.P. c. н.P. D. None of these | 12 |
| 161 | Evalaute the integral ( int_{0}^{1} sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x ) A ( cdot frac{pi}{4}-log 2 ) B ( cdot frac{pi}{2}+log 2 ) c. ( frac{pi}{2}-log 2 ) D. ( frac{pi}{4}+log 2 ) | 12 |
| 162 | The value of ( int e^{x} frac{1+n x^{n-1}-x^{2 n}}{left(1-x^{n}right) sqrt{1-x^{2 n}}} d x ) is equal to A ( cdot e^{x}(sqrt{1-x^{2}})+c ) B. ( e^{x} frac{sqrt{1+x^{2 n}}}{1+x^{2 n}}+c ) C ( frac{e^{x} sqrt{1-x^{n}}}{1-x^{2 n}}+c ) D. ( frac{e^{x} sqrt{1-x^{2 n}}}{1-x^{n}}+c ) | 12 |
| 163 | Integrate: ( int frac{boldsymbol{v}}{1-boldsymbol{v}}= ) | 12 |
| 164 | f(a) 15. If f(x) = 4 ,11 = xg{x(1 – x)}dx iter f(-a) f(a) and 12 = 5 8{x(1 – x)}dx, then the value of , is2004 (a) 1 (b) 3 (c) 1 (d) 2 f(-a) | 12 |
| 165 | Evaluate: ( int frac{cos x-sin x}{1+sin 2 x} d x ) A ( cdot frac{1}{sin x+cos x}+C ) в. ( -frac{1}{sin x+cos x}+C ) c. ( frac{2}{sin 2 x+cos x}+C ) D. ( -frac{2}{sin 2 x+cos x}+C ) | 12 |
| 166 | Assertion ( int_{0}^{pi / 2} x cot x d x=frac{pi}{2} log 2 ) Reason ( int_{0}^{pi / 2} log sin x d x=-frac{pi}{2} log 2 ) A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion B. Both Assertion & Reason are individually true but Reason is not the , correct (proper) explanation of Assertion c. Assertion is true but Reason is false D. Assertion is false but Reason is true | 12 |
| 167 | ( int x sec ^{2} 2 x d x ) A. ( frac{1}{4} x tan 2 x-frac{1}{2} log sec 2 x ) B. ( frac{1}{2} x tan 2 x+frac{1}{4} log sec 2 x ) C ( frac{1}{4} x tan 2 x-frac{1}{4} log sec 2 x ) D. ( frac{1}{2} x tan 2 x+frac{1}{4} log cos 2 x ) | 12 |
| 168 | Evaluate ( int_{0}^{2} frac{6 x+3}{x^{2}+4} d x ) | 12 |
| 169 | If ( I=int_{3}^{5} frac{sqrt{x}}{sqrt{8-x}+sqrt{x}} d x ) then ( I ) equals ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. 3.5 | 12 |
| 170 | Evaluate: ( int_{0}^{pi / 8} cos ^{3} 4 x d x ) A . ( 1 / 6 ) в. ( 1 / 5 ) c. ( -1 / 3 ) D. 1/ | 12 |
| 171 | If ( int sqrt{boldsymbol{x}}left(1-boldsymbol{x}^{3}right)^{-1 / 2} boldsymbol{d} boldsymbol{x}=frac{boldsymbol{2}}{boldsymbol{3}} boldsymbol{g}(boldsymbol{f}(boldsymbol{x}))+boldsymbol{c} ) then A ( cdot f(x)=sqrt{x}, g(x)=sin ^{-1} x ) B . ( f(x)=x^{3 / 2}, g(x)=sin ^{-1} x ) C ( cdot f(x)=x^{2 / 3}, g(x)=cos ^{-1} x ) D. ( f(x)=sqrt{x}, g(x)=cos ^{-1} x ) | 12 |
| 172 | ( int frac{x^{2}-1}{x^{4}+x^{2}+1} d x ) is equal to A ( cdot log left(x^{4}+x^{2}+1right)+c ) B. ( log frac{x^{2}-x+1}{x^{2}+x+1}+c ) ( ^{mathrm{C}} cdot frac{1}{2}^{log frac{x^{2}-x+1}{x^{2}+x+1}}+c ) D. ( frac{1}{2} log frac{x^{2}+x+1}{x^{2}-x+1}+c ) | 12 |
| 173 | Evaluate ( int frac{d x}{(2 x-7) sqrt{(x-3)(x-4)}} ) | 12 |
| 174 | ( int_{0}^{infty} frac{x tan ^{-1} x}{left(1+x^{2}right) x^{2}} d x ) A ( cdot frac{pi}{2} log 2 ) B. c. D. | 12 |
| 175 | Solve ( int frac{v}{1-v} d v ) | 12 |
| 176 | Evaluate: ( int frac{5 x+3}{sqrt{x^{2}+4 x+10}} d x ) | 12 |
| 177 | ( int_{0}^{pi} frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{3}+boldsymbol{2} sin boldsymbol{x}+cos boldsymbol{x}}= ) A . ( pi / 3 ) B . ( pi / 4 ) c. ( pi / 6 ) D . ( pi / 2 ) | 12 |
| 178 | ( int x sqrt{x} d x= ) A ( cdot frac{3}{2} x^{3 / 2}+c ) B ( cdot frac{2}{5} x^{5 / 2}+c ) c. ( frac{5}{2} x^{5 / 2}+c ) D. ( frac{3}{2} sqrt{x}+c ) | 12 |
| 179 | ( frac{boldsymbol{A} boldsymbol{x}-mathbf{1}}{left(mathbf{1}-boldsymbol{x}+boldsymbol{x}^{2}right)(boldsymbol{x}+mathbf{2})}=frac{boldsymbol{x}}{mathbf{1}-boldsymbol{x}+boldsymbol{x}^{2}}- ) ( frac{1}{x+2} Rightarrow A= ) ( A cdot 3 ) B. 2 ( c cdot 4 ) ( D ) | 12 |
| 180 | If ( boldsymbol{M}=int_{0}^{pi / 2} frac{cos boldsymbol{x}}{boldsymbol{x}+mathbf{2}} boldsymbol{d} boldsymbol{x}, boldsymbol{N}= ) ( int_{0}^{pi / 4} frac{sin x cos x}{(x+1)^{2}} d x quad, ) then the value of ( M-N ) is ( ? ) ( A ) в. c. ( frac{2}{pi-4} ) D. ( frac{2}{pi+4} ) | 12 |
| 181 | Integrate the following function. ( sin x sin (cos x) ) | 12 |
| 182 | Evaluate: ( int_{0}^{1} frac{sqrt{tan ^{-1} x}}{1+x^{2}} d x ) | 12 |
| 183 | The value of ( lim _{n rightarrow infty} Sigma_{1}^{n} sin left(frac{pi}{4}+frac{pi i}{2 n}right) frac{pi}{2 n}=? ) ( ^{mathbf{A}} cdot int_{frac{pi}{2}}^{frac{pi}{4}} sin x d x ) B. ( int_{frac{pi}{2}}^{frac{3 pi}{4}} sin x d x ) ( ^{mathrm{c}} cdot int_{frac{pi}{7}}^{frac{3 pi}{4}} sin x d x ) D. ( int_{pi}^{3 pi} sin x d x ) | 12 |
| 184 | ( int frac{d x}{4 sin ^{2} x+4 sin x cos x+5 cos ^{2} x}= ) ( A cdot tan ^{-1}(2 tan x+1)+c ) B. ( tan ^{-1}left(tan x+frac{1}{2}right)+c ) c. ( frac{1}{8} tan ^{-1}left(tan x+frac{1}{2}right)+c ) D. ( frac{1}{4} tan (2 tan x+1)+c ) | 12 |
| 185 | Evaluate: ( int_{0}^{frac{pi}{2}} frac{sin x}{sin x+cos x} d x ) | 12 |
| 186 | ( int frac{log x cdot sin left(1+(log x)^{2}right)}{x} d x= ) A. ( -frac{1}{2} cos left(1+(log x)^{2}right)+c ) B. ( frac{1}{2} cos left(1+(log x)^{2}right)+c ) C. ( frac{1}{2} sin left(1+sin (log x)^{2}right)+c ) D. ( -frac{1}{2} sin left(1+sin (log x)^{2}right)+c ) | 12 |
| 187 | The value of ( int frac{left(a x^{2}-bright) d x}{x sqrt{c^{2} x^{2}-left(a x^{2}+bright)^{2}}} ) is equal to A ( cdot frac{1}{c} sin ^{-1}left(a x+frac{b}{x}right)+k ) B. ( operatorname{csin}^{-1}left(a+frac{b}{x}right)+k ) c. ( sin ^{-1}left(frac{a x+frac{b}{x}}{c}right)+k ) D. none of these | 12 |
| 188 | Assertion The value ( int_{-4}^{-5} sin left(x^{2}-3right) d x+ ) ( int_{-2}^{-1} sin left(x^{2}+12 x+33right) ) is zero Reason ( int_{-a}^{a} f(x) d x=0 ) if ( f(x) ) is an odd function. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 189 | ntegrate the function ( frac{x^{2}+x+1}{(x+1)^{2}(x+2)} ) | 12 |
| 190 | ( int frac{5 x+4}{sqrt{x^{2}+3 x+2}} d x= ) ( A sqrt{x^{2}+3 x+2} ) ( boldsymbol{B} ln left[left(boldsymbol{x}+frac{boldsymbol{3}}{mathbf{2}}right)+sqrt{boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+boldsymbol{2}}right]+boldsymbol{C} ) Then ( A+2 B=? ) ( A cdot 9 ) B . 10 c. 11 D. 12 | 12 |
| 191 | Evaluate the following integrals:( int frac{x+1}{sqrt{x^{2}-x+1}} d x ) | 12 |
| 192 | ( int_{1}^{sqrt{3}} frac{d x}{1+x^{2}} ) | 12 |
| 193 | Solve: ( int frac{1}{(sin x-2 cos x)(2 sin x+cos x)} d x ) | 12 |
| 194 | ( int_{0}^{1} frac{4 x^{3}}{sqrt{1-x^{8}}} d x=? ) ( A cdot pi ) в. ( -pi ) c. ( pi / 2 ) D . ( -pi / 2 ) | 12 |
| 195 | Evaluate the definite integral: ( int_{0}^{pi / 2} cos x d x ) | 12 |
| 196 | Evaluate ( int_{0}^{1} e^{2-3 x} d x ) as a limit of a sum. | 12 |
| 197 | Evaluate the following integral: ( int frac{e^{x}}{sqrt{16-e^{2 x}}} d x ) | 12 |
| 198 | Evaluate ( int frac{(log x)^{2}}{x} d x ) | 12 |
| 199 | Find ( int frac{x+3}{sqrt{5-4 x+x^{2}}} d x ) | 12 |
| 200 | ( mathrm{f} int_{0}^{pi / 2} log (sin mathrm{x}) mathrm{d} mathrm{x}=mathrm{k} ) then ( int_{0}^{pi / 2} log (cos x) d x ) A ( cdot frac{k}{2} ) B . ( 2 k ) ( c .-3 k ) D. | 12 |
| 201 | If ( int frac{(4 x+3)}{sqrt{2 x^{2}+2 x-3}}= ) ( 2 sqrt{2 x^{2}+2 x-3}+ ) ( frac{1}{sqrt{boldsymbol{k}}} log left|boldsymbol{x}+frac{1}{2}+sqrt{boldsymbol{x}^{2}+boldsymbol{x}-frac{boldsymbol{3}}{2}}right|+boldsymbol{C} ) then value of ( k ) is | 12 |
| 202 | ( int frac{cos x}{sqrt[3]{sin ^{2} x}} d x= ) A ( cdot 3 sqrt[3]{sin x}+c ) B. ( 3 sqrt[3]{sin ^{2} x}+c ) c. ( sqrt[3]{sin x}+c ) D. ( sqrt[3]{sin ^{2} x}+c ) | 12 |
| 203 | Evaluate the following integral: ( int frac{sin 2 x}{(a+b cos 2 x)^{2}} d x ) | 12 |
| 204 | The value of ( int frac{d^{2}}{d x^{2}}left(tan ^{-1} xright) d x ) is equal to A ( cdot frac{1}{1+x^{2}}+c ) B. ( tan ^{-1} x+c ) c. ( x tan -frac{1}{2} log left(1+x^{2}right)+C ) D. ( frac{1+x^{2}}{2}+c ) | 12 |
| 205 | ( int frac{d x}{1+x^{3}} ) | 12 |
| 206 | ( mathbf{f} boldsymbol{I}=int boldsymbol{x} sin ^{-1}left{frac{mathbf{1}}{mathbf{2}} sqrt{frac{mathbf{2} boldsymbol{a}-boldsymbol{x}}{boldsymbol{a}}}right} boldsymbol{d} boldsymbol{x}= ) ( frac{boldsymbol{A}}{mathbf{2 4 8}}left(boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{a}^{2}right) cos ^{-1} frac{boldsymbol{x}}{boldsymbol{2} boldsymbol{a}}- ) ( frac{1}{8} x sqrt{4 a^{2}-x^{2}}+C ) then ( A ) is equal to | 12 |
| 207 | ( int x cdotleft(x^{x}right)^{x}(2 log x+1) d x ) ( mathbf{A} cdot x^{left(x^{x}right)}+c ) B. ( left(x^{x}right)^{x}+c ) c. ( x^{x} cdot log x+c ) D. does not exist | 12 |
| 208 | The value of ( int sqrt{2}left(frac{sin x}{sin left(x-frac{pi}{4}right)}right) d x ) is ( ^{mathbf{A}} cdot_{x-log }left|sin left(x-frac{pi}{4}right)right|+c ) B. ( x+log left|cos left(x-frac{pi}{4}right)right|+c ) ( ^{mathbf{C}} x-log left|cos left(x-frac{pi}{4}right)right|+c ) D. ( x+log left|sin left(x-frac{pi}{4}right)right|+c ) | 12 |
| 209 | 7. If [ f(x)dx = v(x), then fx f(x)dx is equal to [JEE M: @ {[r’y(x?)-x?vCx®)dx] +C (1) {x?(3%)=35x?y(x*)dx + c (c) fry(x?)-[xv(x)dx+C (a) }[ry(x?)-[xºv(rº)dx]+C | 12 |
| 210 | Evaluate ( int_{0}^{pi / 4} frac{cos x-sin x}{10+sin 2 x} d x ) A ( cdot frac{1}{3}left(tan ^{-1} frac{sqrt{2}}{3}+tan ^{-1} frac{1}{3}right) ) B. ( frac{1}{3}left(tan ^{-1} frac{sqrt{1}}{3}-cot ^{-1} frac{2}{3}right) ) c. ( frac{1}{3}left(tan ^{-1} frac{sqrt{2}}{3}-tan ^{-1} frac{1}{3}right) ) D. ( frac{1}{3}left(tan ^{-1} frac{sqrt{1}}{3}-cot ^{-1} frac{1}{3}right) ) | 12 |
| 211 | Evaluate ( int frac{1}{left(e^{x}-1right)} d x ) | 12 |
| 212 | Evaluate the given integral: ( int_{0}^{4}left(4 x-x^{2}right) d x ) | 12 |
| 213 | ( int_{-pi / 2}^{pi / 2} cos t cdot sin left(2 t-frac{pi}{4}right) d t= ) A ( frac{sqrt{2}}{3} ) B. ( -frac{sqrt{2}}{3} ) ( c cdot frac{sqrt{3}}{1} ) D. ( frac{1}{sqrt{3}} ) | 12 |
| 214 | Find the integral ( int frac{sin x}{cos ^{2} x} d x ) | 12 |
| 215 | ( int_{0}^{pi} x f(sin x) d x ) is equal to ( mathbf{A} cdot pi int_{0}^{x} f(cos x) d x ) ( mathbf{B} cdot pi int_{0}^{x} f(sin x) d x ) ( ^{mathbf{C}} cdot frac{pi}{2} int_{0}^{x / 2} f(sin x) d x ) D ( cdot pi int_{0}^{pi / 2} f(cos x) d x ) | 12 |
| 216 | Evaluate the following integral: ( int_{0}^{pi / 2} cos x d x ) | 12 |
| 217 | Solve : ( int_{1}^{2} frac{x}{(x+1)(x+2)} d x ) | 12 |
| 218 | 35. Prove that So tan” ( 2) dx = 25. tan! xdx. 1- x + x2) Hence or otherwise, evaluate the integral ſtan-(1 = x + x²) dx. (1998 -8 Marks) | 12 |
| 219 | What is the value of ( int_{-pi / 4}^{pi / 4}(sin x-tan x) d x ) ( ^{mathbf{A}} cdot-frac{1}{sqrt{2}}+ln left(frac{1}{sqrt{2}}right) ) B. ( frac{1}{sqrt{2}} ) c. 0 D. ( sqrt{2} ) | 12 |
| 220 | Evaluate the following definite integral: ( int_{0}^{1}left(3 x^{2}+2 xright) d x ) | 12 |
| 221 | 9. Let f(x) be a function satisfying f'(x)=) 8f'(x)=f(x) with f(0=1 and g(x) be a function that satisfies f(x) + g(x) = satisfies f(x) + g(x) = x2. Then the value of the integral s f(x) g(x)dx, is [2003] nININ (C) e + | 12 |
| 222 | ( int frac{1}{x-2} d x ) | 12 |
| 223 | Evaluate ( int_{1}^{2} frac{-1}{x^{2}} d x ) | 12 |
| 224 | Evaluate ( int frac{e^{-x}}{1+e^{x}} d x ) | 12 |
| 225 | ( int_{0}^{pi} frac{phi d phi}{1+sin phi} ) is equal to A . ( -pi ) в. ( frac{pi}{2} ) c. ( pi ) D. None of these | 12 |
| 226 | c 2. Evaluate / xdx (a+bx) ² | 12 |
| 227 | Integrate: ( int e^{x} sin x cdot d x ) | 12 |
| 228 | Solve: ( int frac{1}{x log x log (log x)} d x ) | 12 |
| 229 | ( frac{1}{(x+1)left(x^{2}+2 x+2right)}=frac{A}{x+1}+ ) ( frac{B x+C}{(x+1)^{2}+1} Rightarrow A+B= ) ( A cdot 2 ) B. – ( c ) ( D ) | 12 |
| 230 | Verify mean value theorem for the function ( f(x)=x^{2} ) in the interval [2,4] | 12 |
| 231 | Evaluate: ( int frac{d x}{sqrt{1-e^{2 x}}} ) | 12 |
| 232 | ( int frac{1}{(x-2)left(x^{2}+1right)} d x= ) A. ( frac{1}{5}left[log |x-2|-frac{1}{2} log left(x^{2}+1right)+2 tan ^{-1} xright]+c ) B. ( frac{1}{5}left[log |x-2|-frac{1}{2} log left(x^{2}+1right)-2 tan ^{-1} xright]+c ) C ( cdot frac{1}{5}left[log |x-2|-frac{1}{2} log left(x^{2}+1right)-2 tan ^{-1} xright]+c ) D. ( -frac{1}{2}left[log |x+2|+frac{1}{2} log left|x^{2}-1right|-2 tan ^{-1} xright]+c ) | 12 |
| 233 | ( int_{-3}^{3} frac{x^{2} sin ^{3} x}{1+x^{8}} d x ) equals ( g(x) ) then ( g(x) ) equal to A . 6 B. 3 ( c cdot 0 ) ( D ) | 12 |
| 234 | Solve: ( int_{0}^{2} x sqrt{x+2} d x ) | 12 |
| 235 | Evalute ( int frac{cot x}{sqrt{sin x}} d x ) | 12 |
| 236 | Evaluate the following integral: ( int_{0}^{5} x^{2} d x ) | 12 |
| 237 | Evaluate the following integral: ( int_{0}^{2}left(x^{2}+2 x+1right) d x ) | 12 |
| 238 | ( int_{0}^{pi} boldsymbol{f}(boldsymbol{x}) boldsymbol{d} boldsymbol{x}= ) A ( cdot frac{4}{pi} ) в. ( frac{8}{pi} ) ( c cdot frac{8}{pi^{2}} ) D. | 12 |
| 239 | ( int_{-}left(3 sin x-4 cos x+5 sec ^{2} x-2 cos e c^{2}right. ) | 12 |
| 240 | ( lim _{n rightarrow infty} frac{left(1^{k}+2^{k}+3^{k}+ldots . .+nright.}{left(1^{2}+2^{2}+ldots . .+n^{2}right)left(1^{3}+2^{3}+right.} ) ( boldsymbol{F}(boldsymbol{k}), ) then ( (boldsymbol{k} in boldsymbol{N}) ) A. ( F(k) ) is finite for ( k leq 6 ) В. ( F(5)=0 ) c. ( F(6)=frac{12}{7} ) D. ( F(6)=frac{5}{7} ) | 12 |
| 241 | Evaluate ( int frac{x}{sqrt{x^{2}+2}} d x ) A ( . I=sqrt{x^{2}-2}+C ) B. ( I=sqrt{x^{2}+2}+C ) c. ( I=sqrt{x^{3}+2}+C ) D. ( I=sqrt{x^{3}-2}+C ) | 12 |
| 242 | Integrate: ( intleft(left(frac{x+1}{x-1}right)^{2}+left(frac{x-1}{x+1}right)^{2}-2right)^{frac{1}{2}} d x ) | 12 |
| 243 | Evaluate: ( int frac{2^{x+1}-5^{x-1}}{10^{x}} d x ) | 12 |
| 244 | ( operatorname{Resolve} frac{x^{4}-x^{2}+1}{x^{2}left(x^{2}+1right)^{2}} ) into partial fractions. A ( cdot-frac{2}{x^{2}}+frac{1}{left(x^{2}+1right)^{2}} ) B. ( -frac{1}{x^{2}}+frac{3}{left(x^{2}+1right)^{2}} ) c. ( frac{1}{x^{2}}-frac{5}{left(x^{2}+1right)^{2}} ) D. ( frac{1}{x^{2}}-frac{3}{left(x^{2}+1right)^{2}} ) | 12 |
| 245 | ( int_{0}^{1} sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x ) | 12 |
| 246 | Integrate the rational function ( frac{2 x}{left(x^{2}+1right)left(x^{2}+3right)} ) | 12 |
| 247 | What is correct about mean value theorem? A. It states that given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. B. It tells us when certain values for the derivative must exist. C. Both A and B D. Only B | 12 |
| 248 | ( lim _{n rightarrow infty}left[frac{1}{n^{2}} sec ^{2} frac{1}{n^{2}}+frac{2}{n^{2}} sec ^{2} frac{4}{n^{2}}+ldots+frac{1}{r}right. ) A ( cdot frac{1}{2} sec 1 ) B ( cdot frac{1}{2} csc 1 ) ( c . tan 1 ) D. ( frac{1}{2} tan ) | 12 |
| 249 | ( int x sin ^{-1} x cdot d x ) | 12 |
| 250 | If ( f(x)=lim _{n rightarrow infty}left[2 x+4 x^{3}+dots dots+right. ) ( left.2 n x^{2 n-1}right](0<x<1) ) then ( int f(x) d x ) is equal to A. ( -sqrt{1-x^{2}}+ ) constant в. ( frac{1}{sqrt{1-x^{2}}}+ ) constant c. ( frac{1}{x^{2}-1}+ ) constant D. ( frac{1}{1-x^{2}}+ ) constant | 12 |
| 251 | (1995) The value of the integral cos x + cos x din x dx is sin? x + sin4 x (a) sin x-6 tan-(sin x) + c (b) sin x -2(sinx)-1 + c sin x -2(sinx) – 6tan (sin x)+c (d) sin x -2(sinx)-1 + 5tan-‘(sin x)+c (c) SIIT | 12 |
| 252 | ( int_{0}^{1} tan ^{-1} x d x ) A. ( frac{pi}{2}-frac{1}{2} log 2 ) B. ( frac{pi}{4}+frac{1}{2} log 2 ) c. ( frac{pi}{2}-frac{1}{4} log 2 . ) D. ( frac{pi}{4}-frac{1}{2} log 2 ) | 12 |
| 253 | Solve ( int_{0}^{pi / 2} frac{cos ^{2} x}{sin ^{2} x+cos ^{2} x} d x ) A. ( -frac{pi}{4} ) в. c. ( frac{3 pi}{4} ) D. None of these | 12 |
| 254 | ( sqrt{2} int_{0}^{2 pi} sqrt{1-sin x} d x= ) | 12 |
| 255 | 98 pk+1 17. 1 k +1 -dxthen x(x+1) krl Jk (JEE Adv. 2017) (b) I log, 99 (c) 1 50 50 | 12 |
| 256 | Integrate the following ( int frac{1}{x^{2}+8 x+20} d x ) A ( cdot frac{1}{2} sin ^{-1} frac{x+4}{2}+C ) B. ( frac{1}{2} cot ^{-1} frac{x+4}{2}+C ) c. ( frac{1}{2} tan ^{-1} frac{x+4}{2}+C ) D. None of these | 12 |
| 257 | ( int frac{2 x+3}{sqrt{4 x+3}} d x= ) A ( cdot frac{1}{12}(4 x-3)^{frac{3}{2}}+frac{1}{4} sqrt{4 x+3}+c ) B ( frac{1}{12}(4 x+3)^{frac{3}{2}}+frac{3}{4} sqrt{4 x+3}+c ) C ( frac{1}{12}(4 x+3)^{frac{3}{2}}-frac{3}{4} sqrt{4 x-3}+c ) D ( quad frac{1}{12}(4 x+3)^{frac{3}{2}}-frac{1}{4} sqrt{4 x-3}+c ) | 12 |
| 258 | 7. * x4(1 – x) dx is (are) The value(s) of 1+12 dx is (are) (c) o | 12 |
| 259 | ( int frac{e^{x}}{x}(1+x cdot ln x) d x ) | 12 |
| 260 | Evaluate ( int frac{(x-1)^{2}}{x^{4}+2 x^{2}+1} d x ) A ( cdot frac{x^{3}}{3}+x+frac{x}{x^{2}+1}+c ) ( ^{text {В } cdot frac{x^{5}+x^{3}+x+3}{3left(x^{2}+1right)}+c} ) c. ( frac{x^{5}+4 x^{3}+3 x+3}{3left(x^{2}+1right)}+c ) D. None of these | 12 |
| 261 | Evaluate: ( int frac{x d x}{(x-1)left(x^{2}+1right)} ) | 12 |
| 262 | ( int_{1}^{e} e^{frac{x^{2}-2}{2}}left(frac{1}{x}+x log xright) d x ) ( int_{1}^{e} e^{x^{2}}-2left(frac{1}{x}+x log xright) d x ) | 12 |
| 263 | State whether the statement is ture/false. ( int_{-pi / 2}^{pi / 2}left(frac{sin x}{1-cos x}right) d x=0 ) A. True B. False | 12 |
| 264 | Integrate the function ( x sin x ) | 12 |
| 265 | 10. The value of the integral I = ( x(1 – x)” dx is +- – n+1 (b) n+2 n+1 (d) n+2 n+1 n +2 | 12 |
| 266 | ( int x sin ^{2} x d x ) A. ( frac{x^{2}}{4}-frac{cos 2 x}{3}+frac{1}{8} sin 2 x+c ) B. ( frac{x^{2}}{4}-frac{x sin 2 x}{4}-frac{1}{8} cos 2 x+c ) C ( frac{x^{2}}{4}+frac{x sin 2 x}{4}+frac{1}{8} cos 2 x+c ) D. ( frac{-x^{2}}{4}-frac{cos 2 x}{3}+frac{1}{8} cos 2 x+c ) | 12 |
| 267 | If ( int frac{x^{2}-x+1}{left(x^{2}+1right)^{frac{3}{2}}} e^{x} d x=e^{x} f(x)+c, ) then This question has multiple correct options ( A . f(x) ) is an even function B. ( f(x) ) is a bounded function c. the range of ( f(x) ) is (0,1] D. f(x) has two points of extrema | 12 |
| 268 | ( int_{0}^{pi} frac{d x}{1+2^{tan x}}= ) ( A cdot O ) B . ( pi / 4 ) c. ( pi / 2 ) D. | 12 |
| 269 | Solve: ( int_{0}^{frac{pi}{2}}(sin 2 x) sin x d x ) | 12 |
| 270 | ( f int frac{2 e^{5 x}+e^{4 x}-4 e^{3 x}+4 e^{2 x}+2 e^{x}}{left(e^{2 x}+4right)left(e^{2 x}-1right)^{2}} d x ) ( =tan ^{-1}left(e^{x / 2}right)-frac{K}{248left(e^{2 x}-1right)}+C ) then ( K ) is equal to | 12 |
| 271 | Evaluate the following integral: ( int_{0}^{3}left(2 x^{2}+3 x+5right) d x ) | 12 |
| 272 | Solve ( int frac{x^{2}}{x+1} d x ) | 12 |
| 273 | Solve ( : int_{0}^{frac{pi}{2}} x^{2} sin x d x ) | 12 |
| 274 | ( int frac{1-x}{1+x} ) | 12 |
| 275 | ( int_{0}^{2 t} frac{f(x)}{f(x)+f(2 t-x)} d x ) A .2 B. 3t ( c cdot t ) D. t/2 | 12 |
| 276 | 22. The value of integra 3 V9-x+ Fax is (C) 2 (d) 1 | 12 |
| 277 | 1923 12. Let f'(x) = sind for all x ER with 165) = 0.18. Ifm< | f(x)dx S M , then the possible values of m and M 1/2 are (JEE Adv. 2015) m=13, M=24 (b) m=,M= (c) m=-11, M=0 (d) m=1,M=12 (a) | 12 |
| 278 | ( int frac{e^{x} d x}{cosh x+sinh x}= ) ( A cdot log cosh x+c ) B. ( tan x+cot x+c ) C ( cdot frac{1}{2} e^{2 x}+c ) D. ( x+c ) | 12 |
| 279 | The value of ( int e^{tan ^{-1} x}left(frac{1+x+x^{2}}{1+x^{2}}right) d x ) is equal to A. ( x e^{tan ^{-1} x}+C ) B. ( x^{2} e^{tan ^{-1} x}+C ) c. ( frac{1}{x} e^{tan ^{-1} x}+C ) D. ( x e^{cot ^{-1} x}+C ) | 12 |
| 280 | Express ( int_{0}^{4} x^{3} d x ) as limit of sum and thus evaluate it. | 12 |
| 281 | Number of Partial Fractions of ( frac{3 x^{2}+1}{left(x^{2}+1right)^{4}} ) A .4 B. 3 ( c cdot 2 ) D. | 12 |
| 282 | Evaluate: ( int e^{x} sin left(e^{x}right) d x ) A ( cdot cos e^{x}+C ) B. – ( cos e^{x}+C ) c. ( left(cos e^{x}right)^{-1}+C ) ( mathbf{D} cdot sin e^{x}+C ) | 12 |
| 283 | Evaluate ( : int frac{1}{sqrt{x^{3}}} d x ) | 12 |
| 284 | Evaluate: ( int frac{cos x+sin x}{sqrt{sin 2 x}} d x ) | 12 |
| 285 | Integrate: ( int frac{x^{2}+1}{(x+1)^{3}(x-2)} d x ) | 12 |
| 286 | Evaluate the following integral: ( int_{0}^{2}left(x^{2}+xright) d x ) | 12 |
| 287 | Evaluate ( : int_{-pi}^{pi} frac{sin ^{2} x}{1+e^{x}} d x ) | 12 |
| 288 | Solve ( int_{boldsymbol{pi} / mathbf{6}}^{boldsymbol{pi} / mathbf{3}} frac{mathbf{1}}{sin mathbf{2} boldsymbol{x}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 289 | 2. 2 sin x -sin 2x f(x) is the integral of – x #0, find lim f'(x) →0 (1979) | 12 |
| 290 | Evaluate ( int x^{2} e^{x} d x= ) A ( cdot e^{x}left(x^{2}-2 x+2right)+c ) B . ( e^{x}left(x^{2}+2 x+2right)+c ) c. ( x^{2}+e x+c ) D. ( e^{x}left(x^{2}+x+2right)+c ) | 12 |
| 291 | Evaluate the definite integral: ( int_{1}^{2} frac{1}{x} d x ) | 12 |
| 292 | Evaluate the given integral. ( int frac{1}{cos x-sin x} d x ) | 12 |
| 293 | Assertion ( int_{-pi / 4}^{pi / 4} x^{3} sin ^{4} x d x neq 0 ) Reason ( int_{-a}^{a} f(x) d x=0 ) if ( f(-x)=-f(x) ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct | 12 |
| 294 | ( frac{1}{boldsymbol{x}^{4}+1}= ) A. ( quadleft[frac{x+sqrt{2}}{2 sqrt{2} sqrt{2}} frac{x+sqrt{2}}{2 sqrt{2}left(x^{2}-sqrt{2} x+1right)}right] ) B. ( quadleft[frac{x+sqrt{2}}{x^{2}+sqrt{2} x+1}-frac{x+sqrt{2}}{left(x^{2}-sqrt{2} x+1right)}right] ) C ( quadleft[frac{x+sqrt{2}}{2 sqrt{2}left(x^{2}+sqrt{2} x+1right)} frac{sqrt{2}-x}{(-sqrt{2})}right] ) D. ( frac{1}{2 sqrt{2}}left[frac{x+sqrt{2}}{left(x^{2}+sqrt{2} x+1right)}+frac{sqrt{2}-x}{left(x^{2}-sqrt{2} x+1right)}right] ) | 12 |
| 295 | If ( int frac{2^{x}}{sqrt{1-4^{x}}} d x=K sin ^{-1}left(2^{x}right)+C ) then ( K ) is equal to A ( . e n 2 ) в. ( frac{1}{2} ell n 2 ) ( c cdot frac{1}{2} ) D. ( frac{1}{ell n^{2}} ) | 12 |
| 296 | Evaluate ( int_{1}^{3}left(2 x^{2}+5 xright) d x ) | 12 |
| 297 | ( int_{0}^{2 pi} cos m x cdot sin n x d x ) where ( m, n ) are integers ( = ) ( A cdot O ) B . ( pi ) ( mathbf{c} cdot pi / 2 ) D. ( 2 pi ) | 12 |
| 298 | Evaluate the following integral: ( int frac{1}{sqrt{1+4 x^{2}}} d x ) | 12 |
| 299 | Evaluate ( int_{0}^{2} 2 x d x ) | 12 |
| 300 | The value of ( int frac{d x}{sqrt{8+3 x-x^{2}}} ) is equal to A ( cdot frac{2}{3} sin ^{-1}left(frac{2 x-3}{sqrt{41}}right)+C ) B. ( frac{3}{2} sin ^{-1}left(frac{2 x-3}{sqrt{41}}right)+C ) c. ( frac{1}{sqrt{41}} sin ^{-1}left(frac{2 x-3}{sqrt{41}}right)+C ) D. ( sin ^{-1}left(frac{2 x-3}{sqrt{41}}right)+C ) | 12 |
| 301 | Evaluate: ( int_{-2}^{3} frac{1}{x+5} d x ) | 12 |
| 302 | Solve ( int x sqrt{x+2} d x ) | 12 |
| 303 | ( frac{3 x-1}{left(1-x+x^{2}right)(2+x)}= ) A ( cdot frac{x}{x^{2}-x+1}-frac{1}{x+2} ) в. ( frac{x}{x^{2}-x+1}+frac{1}{x+2} ) c. ( frac{x}{x^{2}+x+1}+frac{2}{x+2} ) D. ( frac{x}{-x+1}-frac{2}{x+2} ) | 12 |
| 304 | Find ( int frac{2 x}{left(x^{2}+1right)left(x^{4}+4right)} d x ) | 12 |
| 305 | Evaluate ( int frac{2 x}{1+x^{2}} d x ) | 12 |
| 306 | By using the properties of definite integrals, evaluate the integral ( int_{0}^{1} x(1-x)^{n} d x ) | 12 |
| 307 | ( int cot ^{2} x d x= ) | 12 |
| 308 | ( boldsymbol{I}=int boldsymbol{x}^{9} boldsymbol{d} boldsymbol{x} ) | 12 |
| 309 | The value of ( int_{0}^{1} frac{log x}{1+x} d x ) equals This question has multiple correct options A ( cdot frac{alpha^{2}}{12} ) ( ^{mathbf{B}}-int_{0}^{1} frac{log (1+x)}{x} ) ( mathbf{C} cdot-frac{pi^{2}}{12} ) D. None of these | 12 |
| 310 | Evaluate : ( int frac{(1+log x)^{2}}{x} d x ) | 12 |
| 311 | Evaluate : ( int frac{x^{5}}{x^{2}+9} d x ) | 12 |
| 312 | ( int_{0}^{pi / 2} frac{d_{X}}{4 cos ^{2} x+9 sin ^{2} x}= ) A ( cdot frac{pi}{12} ) в. c. D. | 12 |
| 313 | ( int frac{x sin ^{-1} x}{sqrt{1-x^{2}}} d x ) | 12 |
| 314 | ( (A): int e^{x}left(log x+x^{-2}right) d x= ) ( e^{x}left(log x-frac{1}{x}right)+c ) ( (mathrm{R}): int e^{x}left[boldsymbol{f}(boldsymbol{x})+boldsymbol{f}^{prime}(boldsymbol{x})right] boldsymbol{d} boldsymbol{x}=boldsymbol{e}^{boldsymbol{x}} boldsymbol{f}(boldsymbol{x})+boldsymbol{c} ) A. Both A and R are true and R is the correct explanation of B. Both A and R are true but R is not correct explanation of c. ( A ) is true but ( R ) is false D. A is false but R is true | 12 |
| 315 | The function ( f(x)=x^{3}-7 x^{2}+25 x+ ) 8 has exactly roots. ( A cdot 2 ) B. ( c .3 ) D. | 12 |
| 316 | The value of ( lim _{n rightarrow infty} e^{frac{3 i}{n}} cdot frac{3}{n}=? ) A ( cdot e^{4}-1 ) B. ( e^{3}-1 ) c. ( e^{5}-1 ) D. ( e^{3}-2 ) | 12 |
| 317 | Integrate: ( int frac{sec ^{2} sqrt{x}}{sqrt{x}} d x ) ( mathbf{A} cdot I=2 tan sqrt{x}+c ) B. ( I=2 cot sqrt{x}+c ) ( mathbf{c} cdot I=3 tan sqrt{x}+c ) D ( cdot I=2^{2} tan sqrt{x}+c ) | 12 |
| 318 | Solve:- ( int t sqrt{frac{t^{2}+1}{t^{2}-1}} d t ) | 12 |
| 319 | The value of ( int frac{d x}{sqrt{2 x-x^{2}}} ) is ( A cdot sin ^{-1}(x)+c ) B ( cdot sin ^{-1}(x-1)+c ) ( mathbf{c} cdot sin ^{-1}(1+x)+c ) D. ( -sqrt{2 x-x^{2}}+c ) | 12 |
| 320 | Evaluate the given integral. ( int frac{1}{x^{4}-1} d x ) | 12 |
| 321 | Integrate: ( int frac{1}{sqrt{x}+x} d x= ) | 12 |
| 322 | ( int_{0}^{pi / 2} sin 2 x d x ) is A . 2 B. 0 c. 1 D. – – | 12 |
| 323 | ( int_{0}^{pi} x cdot log (sin x) d x= ) A ( cdot pi^{2} log (2) ) В. ( frac{pi^{2}}{2} log (2) ) c. ( frac{pi^{2}}{4} ) D. ( -frac{pi^{2}}{2} log (2) ) | 12 |
| 324 | ( int frac{1}{(2 x+1) sqrt{x^{2}-x-2}} d x= ) A. ( -frac{1}{sqrt{5}} sin ^{-1} frac{7+4 x}{3(2 x+1)}+c ) B. ( -frac{1}{sqrt{5}} cos frac{7+4 x}{3(2 x+1)}+c ) c. ( -frac{1}{sqrt{5}} sinh ^{-1} frac{7+4 x}{3(2 x+1)}+c ) D. ( -frac{1}{sqrt{5}} cosh ^{-1} frac{7+4 x}{3(2 x+1)}+c ) | 12 |
| 325 | Evaluate: ( int frac{x^{2}+1}{x^{4}+1} d x ) equals A. ( frac{1}{sqrt{2}} tan ^{-1}left(frac{x^{2}-1}{sqrt{2} x}right)+C ) B. ( frac{1}{sqrt{2}} tan ^{-1}left(frac{1-x^{2}}{sqrt{2} x}right)+C ) c. ( frac{1}{2} tan ^{-1}left(frac{x^{2}-1}{sqrt{2} x}right)+C ) D. ( frac{1}{2} tan ^{-1}left(frac{1-x^{2}}{sqrt{2} x}right)+C ) | 12 |
| 326 | Find the integrals of the functions in Exercises 1 to 22 1. ( sin ^{3}(2 x+1) ) 2. ( sin ^{3} x cos ^{3} x ) ( 3 cdot frac{cos x-sin x}{1+sin 2 x} ) | 12 |
| 327 | 11-COS Mx 29. Let Im = J 1- cos x dx . Use mathematical induction to 0 COS X prove that Im = m,m=0, 1, 2, …… (1995 – 5 Marks) | 12 |
| 328 | Solve ( int(3 x-2) sqrt{2 x^{2}-x+1} d x ) | 12 |
| 329 | Solve: ( int_{0}^{pi / 4} frac{sin x cos x}{cos ^{4} x+sin ^{4} x} d x ) | 12 |
| 330 | Evaluate the integral ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x ) | 12 |
| 331 | The mean value of the function ( boldsymbol{f}(boldsymbol{x})= ) ( frac{2}{e^{x}+1} ) on the interval [0,2] is ( ^{A} cdot_{2-log _{e}}left(frac{2}{e^{2}+1}right) ) B. ( _{2+log _{e}}left(frac{2}{e^{2}+1}right) ) ( ^{mathrm{c}} 2+log _{e}left(frac{2}{e^{2}-1}right) ) D. ( _{-2+log _{e}}left(frac{2}{e^{2}-1}right) ) | 12 |
| 332 | Find Integrals of given function: ( int tan theta tan ^{2} theta sec ^{2} theta d theta ) ( ^{A} cdot frac{2 tan ^{4} theta}{4}+c ) ( ^{text {В }} cdot frac{tan ^{4} theta}{4}+c ) c. ( frac{tan ^{4} theta}{8}+c ) D. None of these | 12 |
| 333 | ( sinh ^{-1}left(frac{x}{4}right) d x ) is equal to A ( cdot x sinh ^{-1}left(frac{x}{4}right)-sqrt{x^{2}+16}+c ) B. ( x sinh ^{-1}left(frac{x}{4}right)+sqrt{x^{2}+16}+c ) c. ( x sinh ^{-1}left(frac{x}{4}right)-frac{1}{2} sqrt{x^{2}+16}+c ) D ( x sinh ^{-1}left(frac{x}{2}right)-x sqrt{x^{2}+16}+c ) | 12 |
| 334 | Solve : ( int frac{x d x}{left(x^{2}+a^{2}right)left(x^{2}+b^{2}right)} ) | 12 |
| 335 | Evaluate the following integral: ( int frac{sin ^{2} x}{1+cos x} d x ) | 12 |
| 336 | ( int cos x log left(tan frac{x}{2}right) d x= ) ( A cdot sin x log |tan x|-x+c ) B. ( -sin x log left|tan frac{x}{2}right|+x+c ) ( mathbf{c} cdot-sin x log left|tan frac{x}{2}right|-x+c ) ( mathbf{D} cdot sin x log left|tan frac{x}{2}right|-x+c ) | 12 |
| 337 | Evaluate the integral ( int_{0}^{pi} x sin ^{5} x cos ^{6} x d x=? ) A ( cdot frac{5 pi}{16} ) в. ( frac{35 pi}{128} ) c. ( frac{5 pi}{8} ) D. ( frac{8 pi}{693} ) | 12 |
| 338 | Evaluate ( int_{0}^{6}(x+2) d x ) | 12 |
| 339 | ( int frac{1}{sqrt{x}} tan ^{4} sqrt{x} sec ^{2} sqrt{x} d x= ) A ( cdot 2 tan ^{5} sqrt{x}+c ) B. ( frac{1}{5} tan ^{5} sqrt{x}+c ) c. ( frac{2}{5} tan ^{5} sqrt{x}+c ) D. None of these | 12 |
| 340 | Integrals of sum particular function prove that ( int frac{d x}{x^{2}-a^{2}}=frac{1}{2 a} log left|frac{x-a}{x+a}right|+c ) | 12 |
| 341 | ( frac{x^{2}+1}{left(x^{2}+2right)left(2 x^{2}+1right)}= ) ( kleft[frac{1}{x^{2}+2}+frac{1}{2 x^{2}+1}right] Rightarrow k= ) ( A cdot frac{1}{4} ) B. 3 ( c cdot frac{1}{5} ) D. | 12 |
| 342 | Solve: ( int frac{1}{sqrt{1-e^{2 x}}} d x ) | 12 |
| 343 | Evaluate : ( int frac{x^{2}-1}{(x-1)^{2}(x+3)} d x ) | 12 |
| 344 | Evaluate ( int_{0}^{1}left(x+x^{2}right) d x ) | 12 |
| 345 | JL 1/2 20. The integral dx equal to (2002) -1/2″ (a) (b) 0 (0) 1 (d) 2en(1) | 12 |
| 346 | ( 4 int frac{a^{6}+x^{8}}{x} d x ) is equal to B. ( a^{6} ln frac{sqrt{a^{6}+x^{8}}-a^{3}}{sqrt{a^{6}+x^{8}}+a^{3}}+c ) ( I=sqrt{a^{6}+x^{8}}+frac{a^{3}}{2} ln left|frac{sqrt{a^{6}+x^{8}}-a^{3}}{sqrt{a^{6}+x^{8}}+a^{3}}right|+c ) ( ^{mathrm{D}} a^{6} ln frac{sqrt{a^{6}+x^{8}}+a^{3}}{sqrt{a^{6}+x^{8}}-a^{3}}+c ) | 12 |
| 347 | ( int_{1}^{2} x^{2} log x d x_{=} ) A. ( frac{8}{3} log 2-frac{7}{9} ) в. ( frac{8}{3} log 2+frac{7}{9} ) c. ( frac{8}{3} log frac{1}{3}-frac{7}{9} ) D. ( frac{8}{3} log frac{1}{3}+frac{7}{9} ) | 12 |
| 348 | ( int cos left{2 tan ^{-1} sqrt{frac{1-x}{1+x}}right} d x ) is equal to A ( cdot frac{1}{8}left(x^{2}-1right)+k ) B. ( frac{1}{2} x^{2}+k ) c. ( frac{1}{2} x+k ) D. none of these | 12 |
| 349 | Show that ( int_{0}^{frac{pi}{2}}[2 log sin x- ) ( log (sin 2 x)] d x=frac{pi}{2} log _{e}left(frac{1}{2}right) ) | 12 |
| 350 | ( int frac{1}{e^{x}+e^{-x}} d x ) ( A cdot tan ^{-1} x ) B. ( tan ^{-1} e^{x} ) ( mathbf{c} cdot cot ^{-1} e^{x} ) D. ( frac{1}{2} cot ^{-1} e^{x} ) | 12 |
| 351 | Evaluate the definite integral ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{1-boldsymbol{x}^{2}}} ) | 12 |
| 352 | Prove that ( boldsymbol{I}= ) ( int_{0}^{frac{pi}{2}} frac{sqrt{sec x}}{sqrt{operatorname{cosec} x}+sqrt{sec x}} d x=frac{pi}{4} ) | 12 |
| 353 | ( int_{0}^{pi / 2} frac{cos x}{1+sin x} d x= ) ( A cdot log 2 ) B. ( log ) e ( c cdot frac{1}{2} log 3 ) D. | 12 |
| 354 | X + n X + X + 13. Let f(x) = lim , for n +00 * *(**»*)(***)-(x+3) all x > 0. Then (JEE Adv. 2016) (c) f'(2)<0. f'(3) f'(2) f(3) f(2) (d) | 12 |
| 355 | Evaluate: ( int sin ^{4} x cos ^{4} x d x ) | 12 |
| 356 | Evaluate ( int_{0}^{2} frac{x}{3} d x ) | 12 |
| 357 | Solve : ( int x^{2}left(1-frac{1}{x^{2}}right) d x ) | 12 |
| 358 | Evaluate the following integral: ( int_{0}^{pi} 1+sin x d x ) | 12 |
| 359 | ble function Let f (x) be a non-constant twice differentiable fun definied on (-00,00) such that f (x) = f (1 – x) an (2008) f = 0. Then, (a) F”(x) vanishes at least twice on [0, 1] (6) Fe=0 @ () sinxd = 0 -1/2 1/2 (d) f(1 – t) esin at dt f(t) esin ni dt = 1/2 | 12 |
| 360 | Evaluate ( int frac{e^{x-1}+x^{e-1}}{e^{x}+x^{e}} d x ) | 12 |
| 361 | Solve: ( int frac{1}{1+x^{4}} d x ) A ( cdot frac{1}{4 sqrt{2}} log left(frac{x^{2}+sqrt{2} x-1}{x^{2}-sqrt{2} x+1}right)+frac{1}{4 sqrt{2}} tan ^{-1}left(frac{sqrt{2} x}{1-x^{2}}right)+C ) B ( cdot frac{1}{sqrt{2}} log left(frac{x^{2}+sqrt{2} x+1}{x^{2}-sqrt{2} x+1}right)-frac{1}{2 sqrt{2}} tan ^{-1}left(frac{sqrt{2} x}{1-x^{2}}right)+C ) C ( frac{1}{2 sqrt{2}} tan ^{1} frac{x^{2}-1}{sqrt{2} x}+frac{1}{4 sqrt{2}} log left|frac{x^{2}+1+sqrt{2} x}{x^{2}+1-sqrt{2} x}right|+c ) D. ( frac{1}{2 sqrt{2}} log left(frac{x^{2}+sqrt{2} x-1}{x^{2}-sqrt{2} x+1}right)-frac{1}{sqrt{2}} tan ^{-1}left(frac{sqrt{2} x}{1-x^{2}}right)+C ) | 12 |
| 362 | ( int frac{sec ^{8} x}{cos e c x} d x= ) A ( cdot frac{cos ^{7} x}{7}+c ) в. ( frac{7}{cos ^{7} x}+c ) c. ( frac{1}{7 cos ^{7} x}+c ) D. ( frac{1}{cos ^{7} x}+c ) | 12 |
| 363 | ( int frac{d x}{xleft(x^{2}+1right)^{2}}= ) A. ( ln frac{|x|}{sqrt{x^{2}+1}}+frac{1}{2left(x^{2}+1right)}+K ) B. ( ln frac{|x|}{sqrt{x^{2}+1}}-frac{3}{2left(x^{2}+1right)}+K ) C. ( -ln frac{|x|}{sqrt{x^{2}+1}}+frac{3}{2left(x^{2}+1right)}+K ) D. ( -ln frac{|x|}{sqrt{x^{2}+1}}+frac{3}{2(x+1)}+K ) | 12 |
| 364 | Find ( int frac{sin x}{sin 4 x} d x ) | 12 |
| 365 | 47. The integral [JEE – is equal to: 1+ cos x (b) -2 (d) 4 (a) (c) -1 2 | 12 |
| 366 | Evaluate: ( int frac{e^{x}}{e^{x}+1} d x ) | 12 |
| 367 | ( n stackrel{L t}{rightarrow} infty frac{1}{n}left{sin ^{2} frac{pi}{2 n}+sin ^{2} frac{2 pi}{2 n}+ldots+right. ) ( left.sin ^{2} frac{n pi}{2 n}right}= ) A. в. ( c cdot 1 / 2 ) D. | 12 |
| 368 | ( int frac{cos ^{2} x}{2+sin x} d x ) | 12 |
| 369 | ( int_{0}^{1}(sqrt{boldsymbol{x}}) boldsymbol{d} boldsymbol{x}+int_{0}^{4 / 3}(sqrt{boldsymbol{4}-boldsymbol{3} boldsymbol{x}}) d boldsymbol{x} ) | 12 |
| 370 | Integrate: ( int cos x log cos x d x ) | 12 |
| 371 | ( int frac{sin x}{1+cos ^{2} x} d x ) | 12 |
| 372 | ( int e^{x}left(tan x+sec ^{2} xright) d x ) | 12 |
| 373 | ( int frac{x^{3}-1}{x^{3}+x} d x ) is equal to ( mathbf{A} cdot x-log x+log left(x^{2}+1right)-tan ^{-1} x+c ) B. ( x-log x+frac{1}{2} log left(x^{2}+1right)-tan ^{-1} x+c ) c. ( x+log x+frac{1}{2} log left(x^{2}+1right)+tan ^{-1} x+c ) D ( x+log x-frac{1}{2} log left(x^{2}+1right)-tan ^{-1} x+c ) | 12 |
| 374 | 21. The value of -dx , a > 0, is [2005] 1+ ax 21. The value of I cos * dx , a>0, is (2) an (6) 7 (c) To – T (2005) (d) 21 (a) an (d) 211 | 12 |
| 375 | If ( boldsymbol{I}=int_{0}^{a} sqrt{frac{boldsymbol{a}-boldsymbol{x}}{boldsymbol{a}+boldsymbol{x}}} boldsymbol{d} boldsymbol{x}, boldsymbol{a}>mathbf{0}, ) then ( boldsymbol{I} ) equals A ( cdot frac{1}{2}left(a-frac{pi}{2}right) ) B ( cdot frac{a}{2}(pi-1) ) c. ( frac{1}{sqrt{2}} a(pi-1) ) D. ( aleft(frac{pi}{2}-1right) ) | 12 |
| 376 | The value of ( int_{0}^{pi / 2} x(sqrt{tan x}+sqrt{cot x}) d x ) is? ( mathbf{A} cdot frac{pi}{2 sqrt{2}} ) B. ( frac{pi^{2}}{2} ) ( frac{pi^{2}}{2 sqrt{2}} ) D. ( frac{pi^{2}}{2 sqrt{3}} ) | 12 |
| 377 | Evaluate: ( int frac{cos 2 x-cos 2 alpha}{cos x-cos alpha} d x ) | 12 |
| 378 | Show that ( int a^{x} e^{x} d x=frac{a^{x} e^{x}}{log a+1} ) | 12 |
| 379 | let ( boldsymbol{f}(boldsymbol{theta})=frac{1}{1+(tan theta)^{2013}} ) then value of ( sum_{theta=1^{0}}^{89^{circ}} f(theta) ) equals A . 45 B. 44 c. ( 89 / 2 ) D. ( 91 / 2 ) | 12 |
| 380 | Solve ( int frac{cos (x+a)}{cos (x-a)} d x ) | 12 |
| 381 | Find the value of definite integrals as the limit of a sum (by first principle). ( int_{a}^{b} e^{-x} d x ) | 12 |
| 382 | ( int frac{(2 x+1)}{(x+2)(x-3)} d x ) | 12 |
| 383 | Evaluate ( int frac{boldsymbol{d x}}{sqrt{mathbf{2 a x}-boldsymbol{x}^{2}}} ) | 12 |
| 384 | Solve ( intleft(4 e^{3 x}+1right) d x ) | 12 |
| 385 | The value of integral ( int_{pi / 4}^{pi / 2} cos x d x ) is? | 12 |
| 386 | 16. The value of I completa dir, >0, is (2015) (a) a (6) at an (c) T2 (d) 20 | 12 |
| 387 | ( intleft(frac{4 e^{x}-25}{2 e^{x}-5}right) d x=A x+ ) ( boldsymbol{B} log left|mathbf{2} e^{x}-mathbf{5}right|+boldsymbol{c}, ) then A. ( A=5, B=3 ) В. ( A=5, B=-3 ) ( mathbf{c} cdot A=-5, B=3 ) D. ( A=-5, B=-3 ) | 12 |
| 388 | ( int_{2}^{3} frac{(x+2)^{2}}{2 x^{2}-10 x+53} d x ) is equal to ( A cdot 2 ) B. ( c cdot 1 / 2 ) D. ( 5 / 2 ) | 12 |
| 389 | Assertion ( int_{-1}^{1} frac{sin x-x^{4}}{4-|x|} d x ) is same as ( int_{0}^{1} frac{-2 x^{4}}{4-|x|} d x ) Reason ( int_{-1}^{1}(f(x)+g(x)) d x=2 int_{0}^{1} f(x) d x ) if ( g(x) ) is an odd function and ( f(x) ) is an even function. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 390 | ( int_{-1}^{1} sqrt{left(frac{x+2}{x-2}right)^{2}}+left(frac{x-2}{x+2}right)^{2}-2 d x ) A ( cdot sin frac{4}{3} ) B. ( operatorname{sln} frac{3}{4} ) c. ( 4 ln frac{4}{3} ) ( D ) | 12 |
| 391 | The value of ( int x(operatorname{cosec} x cot x) d x= ) is A. ( x operatorname{cosec} x-log |tan x / 2|+c ) B. ( 2-x operatorname{cosec} x+log left|tan frac{x}{2}right|+c ) c. ( x operatorname{cosec} x-2 log |tan x / 2|+c ) D. ( x ) cot ( x-log left|tan frac{x}{2}right|+c ) | 12 |
| 392 | Solve : ( int_{0}^{pi} frac{1}{a^{2}-2 a cos x+1} d x ) | 12 |
| 393 | ( int frac{sqrt{cot x}-sqrt{tan x}}{sqrt{2}(cos x+sin x)} d x ) equals to A ( cdot sec ^{-1}(sin x+cos x)+c ) B. ( sec ^{-1}(sin x-cos x)+c ) c. ( ln |(sin x+cos x)+sqrt{sin 2 x}|+c ) D. ( ln |(sin x-cos x)+sqrt{sin 2 x}|+c ) | 12 |
| 394 | Solve: ( int log left(1+x^{2}right) d x ) A ( cdot x log left(1+x^{2}right)-2 x+2 tan ^{-1} x+c ) B ( cdot x log left(1+x^{2}right)-2 x-2 tan ^{-1} x+c ) C ( cdot log left(1+x^{2}right)-2 x+2 tan ^{-1} x+c ) D. ( log left(1+x^{2}right)+2 x-2 tan ^{-1} x+c ) | 12 |
| 395 | Evaluate the definite integral: ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x ) | 12 |
| 396 | If ( int frac{cos 4 x+1}{cot x-tan x} d x=A cos 4 x+B ) where ( A & B ) are constants, then A ( . A=-1 / 4 & B ) may have any value B. ( A=-1 / 8 & B ) may have any value c. ( A=-1 / 2 & B=-1 / 4 ) D. ( A=B=1 / 2 ) | 12 |
| 397 | Integrate with respect to ( x ) ( x ln x ) | 12 |
| 398 | ( int_{0}^{pi / 2} frac{d x}{sin x} ) equals ( mathbf{A} cdot mathbf{0} ) B. ( frac{1}{2} ) ( c cdot 1 ) D. ( 3 / 2 ) | 12 |
| 399 | Let ( f(x) ) be the function part of the integral part, the find ( f(0) ) ( int frac{e^{x}left(x^{3}+x+1right)}{left(x^{2}+1right)^{3 / 2}} d x ) | 12 |
| 400 | ( frac{1}{a^{2}-x^{2}}= ) A ( cdot frac{1}{a(a-x)}+frac{1}{2 a(a+x)} ) s. ( frac{1}{3 a(a-x)}+frac{1}{2 a(a+x)} ) c. ( frac{1}{2 a(a-x)}+frac{1}{2 a(a+x)} ) D. ( frac{1}{2 a(a-x)}+frac{1}{a(a+x)} ) | 12 |
| 401 | Integrate ( int e^{x}left(frac{x^{2}+3 x+3}{(x+2)^{2}}right) d x ) | 12 |
| 402 | Evaluate the following integral ( int frac{x sin ^{-1} x^{2}}{sqrt{1-x^{4}}} d x ) | 12 |
| 403 | By using the properties of definite integrals, evaluate the integral ( int_{0}^{frac{pi}{2}} frac{sqrt{sin x}}{sqrt{sin x}+sqrt{cos x}} d x ) | 12 |
| 404 | If ( int frac{x^{5} d x}{sqrt{1+x^{3}}}=frac{2}{9} sqrt{a+x^{3}}left(x^{3}-bright)+C ) then the value of ( 2 a+3 b ) is equal to | 12 |
| 405 | Solve: ( int cos ^{2} x sin ^{2} x d x ) | 12 |
| 406 | ( boldsymbol{x}^{5} sqrt{boldsymbol{a}^{3}+boldsymbol{x}^{3}} ) | 12 |
| 407 | Evaluate the following: ( int frac{1}{sqrt{x^{2}+4 x+29}} d x ) | 12 |
| 408 | ( sqrt{x^{2}+2 x+5} d x ) is equal to ( mathbf{A} cdot(x+1) sqrt{x^{2}+2 x+5}+frac{1}{2} log |x+1+sqrt{x^{2}+2 x+5}|+C ) В . ( (x+1) sqrt{x^{2}+2 x+5}+2 log |x+1+sqrt{x^{2}+2 x+5}|+C ) c. ( (x+1) sqrt{x^{2}+2 x+5-2} log |x+1+sqrt{x^{2}+2 x+5}|+C ) D ( cdot frac{1}{2}(x+1) sqrt{x^{2}+2 x+5}+2 log |x+1+sqrt{x^{2}+2 x+5}|+ ) | 12 |
| 409 | ( int_{0}^{pi / 4} frac{tan ^{3} x}{1+cos 2 x} d x ) | 12 |
| 410 | ( operatorname{Let} frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{F}(boldsymbol{x})=frac{boldsymbol{e}^{boldsymbol{s} boldsymbol{m} boldsymbol{x}}}{boldsymbol{x}}, boldsymbol{x}>mathbf{0} ) ( int_{1}^{4} frac{mathbf{3}}{boldsymbol{x}} boldsymbol{e}^{s boldsymbol{m} boldsymbol{x}^{3}} boldsymbol{d} boldsymbol{x}=boldsymbol{F}(boldsymbol{k})-boldsymbol{F}(1), ) then one of the possible values of ( k ) is A . 16 B. 62 c. 64 D. 15 | 12 |
| 411 | Evaluate ( int frac{tan ^{7} sqrt{x} sec ^{2} sqrt{x}}{sqrt{x}} d x ) | 12 |
| 412 | ( int frac{cos 2 x}{sin x} d x ) | 12 |
| 413 | ( operatorname{Let} u=int_{0}^{infty} frac{d x}{x^{4}+7 x^{2}+1} & v= ) ( int_{0}^{infty} frac{x^{2} d x}{x^{4}+7 x^{2}+1} ) then: This question has multiple correct options ( A cdot v>u ) B. ( 6 v=pi ) c. ( 3 u+2 v=5 pi / 6 ) D. ( u+v=pi / 3 ) | 12 |
| 414 | Evaluate the given integral. ( int e^{x}(sec x(1+tan x)) d x ) | 12 |
| 415 | The value of the integer ( int_{0}^{pi} e^{cos ^{2} x} cdot cos ^{3}(2 n+1) x d x, n ) integer is ( mathbf{A} cdot mathbf{0} ) B. c. ( 2 pi ) D. none of these | 12 |
| 416 | ( int frac{boldsymbol{d} boldsymbol{x}}{(boldsymbol{x}+mathbf{1})(boldsymbol{x}-mathbf{2})}=boldsymbol{A} log (boldsymbol{x}+mathbf{1})+ ) ( boldsymbol{B} log (boldsymbol{x}-boldsymbol{2})+boldsymbol{C}, ) where This question has multiple correct options ( mathbf{A} cdot A+B=0 ) в. ( A B=0 ) c. ( frac{A}{B}=-1 ) D. none of these | 12 |
| 417 | ( int frac{3 x+1}{left(x^{3}-x^{2}-x+1right)} d x ) | 12 |
| 418 | Evaluate the following integral as limit of sum: ( int_{0}^{2} e^{x} d x ) | 12 |
| 419 | If ( int_{0}^{1} frac{tan ^{-1} x}{x} d x ) is equal to A ( cdot int_{0}^{frac{pi}{2}} frac{sin x}{x} d x ) B ( cdot int_{0}^{frac{pi}{2}} frac{x}{sin x} d x ) ( ^{mathbf{C}} cdot frac{1}{2} int_{0}^{frac{pi}{2}} frac{sin x}{x} d x ) ( ^{mathrm{D}} cdot frac{1}{2} int_{0}^{frac{pi}{2}} frac{x}{sin x} d x ) | 12 |
| 420 | Let ( f(x)=x^{3}-16 x ) and let ( c ) be the number that satisfies the Mean value theorem for ( f ) on the interval [-4,2] What is ( c ) ? A . -1 B . 2 ( c cdot 0 ) D. – | 12 |
| 421 | ( frac{boldsymbol{x}+mathbf{1}}{(mathbf{2} boldsymbol{x}-mathbf{1})(mathbf{3} boldsymbol{x}+mathbf{1})}=frac{boldsymbol{A}}{mathbf{2} boldsymbol{x}-mathbf{1}}+ ) ( frac{B}{3 x+1} Rightarrow 16 A+9 B= ) ( A ) B. 5 ( c cdot 6 ) ( D ) | 12 |
| 422 | ( f frac{1-cos x}{cos x(1+cos x)}=frac{sin alpha}{cos x}-frac{2}{1+cos x} ) then ( boldsymbol{alpha}= ) A. ( frac{pi}{8} ) B. ( c cdot frac{pi}{2} ) ( D ) | 12 |
| 423 | Find: ( int frac{e^{x} d x}{left(e^{x}-1right)^{2}left(e^{x}+2right)} ) | 12 |
| 424 | ( int frac{7^{2 x+3} sin ^{2} 2 x+cos ^{2} 2 x}{sin ^{2} 2 x}=frac{7^{2 x+3}}{2 log 7}- ) ( frac{(cot x+x)}{b} cdot ) Find ( b ) | 12 |
| 425 | Evaluate the following definite integrals ( int_{0}^{3} x^{2} d x ) | 12 |
| 426 | ( intleft(e^{a log x}+e^{x log a}right) d x ) | 12 |
| 427 | If ( int_{0}^{1} cot ^{-1}left(1-x+x^{2}right) d x= ) ( lambda int_{0}^{1} tan ^{-1} x d x, ) then ( lambda ) is equal to ( A ) B. 2 ( c .3 ) ( D ) | 12 |
| 428 | Evaluate the following integrals: ( int frac{1}{x^{2 / 3} sqrt{x^{2 / 3}-4}} d x ) | 12 |
| 429 | Evaluate the integral ( int_{0}^{1} sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x ) using substitution. | 12 |
| 430 | ( int_{0}^{a} x^{4}left(a^{2}-x^{2}right)^{1 / 2} d x ) equals ( ^{A} cdot frac{pi a^{5}}{32} ) в. ( frac{pi a^{6}}{32} ) c. ( frac{pi a^{2}}{32} ) D. None of these | 12 |
| 431 | Prove that: ( int tan ^{3} 2 x sec 2 x d x ) | 12 |
| 432 | Evaluate ( int frac{log (x / e)}{(log x)^{2}} d x ) A ( cdot frac{log x}{x}+c ) B. ( frac{x}{log x}+c ) c. ( frac{x}{log (x)^{2}}+c ) D. ( frac{(log x)^{2}}{x}+c ) | 12 |
| 433 | ( int frac{1}{1+sin x} d x= ) A ( cdot tan x+sec x+c ) B. ( tan x-sec x+c ) c. ( cot x-operatorname{cosec} x+c ) D. – ( cot x+sec x+c ) | 12 |
| 434 | Evaluate the given integral. ( int frac{sec ^{2} sqrt{x}}{sqrt{x}} d x ) | 12 |
| 435 | ( int(2+log x)(e x)^{x} d x=dots .+C ; x>1 ) ( mathbf{A} cdot(e x)^{x} ) B . ( x^{x} ) c. ( (e x)^{-x} ) D. ( e^{x^{x}} ) | 12 |
| 436 | Find ( int_{0}^{5}(x+1) d x ) as limit of a sum | 12 |
| 437 | Let ( f(x)=x^{3}-6 x^{2}-10 x ) and let ( c ) be the number that satisfies the Mean value theorem for ( f ) on the interval ( [-4,5] . ) What is ( c ? ) A . -1 B. -2 c. 0 D. | 12 |
| 438 | ( int_{pi / 6}^{pi / 3} frac{sin ^{3} x}{sin ^{3} x+cos ^{3} x} d x= ) A ( cdot frac{pi}{2} ) в. ( c cdot frac{pi}{12} ) D. ( frac{pi}{6} ) | 12 |
| 439 | et 2x(1+ sin x) dx is [20021 21 + cos²x 1+ cos²x (a) Te? (b) ? (c) zero Zero (d) como yond yaxe | 12 |
| 440 | ( lim _{n rightarrow infty}left[frac{1}{n^{2}} sec ^{2} frac{1}{n^{2}}+frac{2}{n^{2}} sec ^{2} frac{4}{n^{2}}+frac{3}{n^{2}} sec ^{2}right. ) equals A ( cdot frac{1}{2} operatorname{cosec} 1 ) B. ( frac{1}{2} ) sec c. ( frac{1}{2} tan 1 ) D. ( tan 1 ) | 12 |
| 441 | Solve : ( int frac{e^{x}}{-2left(1+e^{-x}right)^{2}} cdot d x ) | 12 |
| 442 | Solve it ( mathbf{2} boldsymbol{I}=int_{boldsymbol{O}}^{boldsymbol{Q}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 443 | ( int_{2}^{4} frac{sqrt{x^{2}-4}}{x^{4}} d x= ) ( A cdot frac{3}{32} ) B. ( frac{sqrt{3}}{32} ) ( c cdot 3 ) 8 D. ( frac{sqrt{3}}{8} ) | 12 |
| 444 | Evaluate: ( int frac{1}{sin x+sec x} d x ) | 12 |
| 445 | Integrate ( int e^{sin x} cdot cos x d x ) | 12 |
| 446 | Solve ( int tan ^{-1} sqrt{x} d x= ) | 12 |
| 447 | ( int_{0}^{pi / 2} frac{cos x-sin x}{1+cos x sin x} d x ) is equal to: A. в. ( frac{pi}{2} ) c. D. ( frac{pi}{6} ) | 12 |
| 448 | ( int_{0}^{1} frac{2 sin ^{-1} frac{x}{2}}{x} d x ) is equal to ( ^{mathrm{A}} cdot int_{0}^{pi / 6} frac{x}{tan x} d x ) в. ( int_{0}^{pi / 6} frac{2 x}{tan x} d x ) ( ^{mathrm{C}} int_{0}^{pi / 2} frac{2 x}{tan x} d x ) D. None of these | 12 |
| 449 | Integrate 🙁 int frac{d x}{x(x+1)} ) | 12 |
| 450 | Evaluate the integral ( int_{-2 pi}^{2 pi} sin ^{5} x d x ) A ( cdot frac{pi^{2}}{2} ) в. ( frac{pi}{15} ) c. ( frac{pi}{17} ) D. 0 | 12 |
| 451 | ( lim _{x rightarrow 0} frac{int_{0}^{x}left(t^{2}+e^{t^{2}}right)^{frac{1}{1-cos t}}}{left(e^{x}-1right)} d t ) is equal to A ( cdot e^{4} ) B ( cdot e^{2} ) ( c cdot e^{3} ) ( D ) | 12 |
| 452 | If ( I_{n}=int_{0}^{pi / 4} tan n x d x ) then ( lim _{n rightarrow infty} nleft(I_{n}+I_{n-2}right)= ) ( A ) B. 1/2 ( c cdot alpha ) D. | 12 |
| 453 | ( int e^{x}left(sec ^{2} x+tan xright) d x ) | 12 |
| 454 | If ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{e}^{boldsymbol{x}}}{mathbf{1}+boldsymbol{e}^{boldsymbol{x}}}, boldsymbol{A}=int_{boldsymbol{f}(-boldsymbol{a})}^{f(boldsymbol{a})} boldsymbol{x} boldsymbol{g}{boldsymbol{x}(mathbf{1}- ) ( boldsymbol{x})} boldsymbol{d} boldsymbol{x} ) and ( boldsymbol{B}=int_{boldsymbol{f}(-boldsymbol{a})}^{f(boldsymbol{a})} boldsymbol{g}{boldsymbol{x}(boldsymbol{1}-boldsymbol{x})} boldsymbol{d} boldsymbol{x} ) then ( frac{B}{A} ) is equal to ( A cdot-1 ) B . – – ( c cdot 2 ) ( D ) | 12 |
| 455 | ( int frac{(x-1)^{2}}{x^{4}+x^{2}+1}= ) | 12 |
| 456 | Evaluate ( int frac{10 n^{9}+10^{n} ln 10}{sqrt{n^{10}+10^{n}+10^{10}}} d n ) | 12 |
| 457 | Evaluate the given integral. ( int frac{sqrt{1-cos 2 x}}{2} d x ) | 12 |
| 458 | ( int_{-pi}^{pi} sin m x sin n x d x=? ) | 12 |
| 459 | Evaluate ( int_{0}^{sqrt{3}} frac{1}{1+x^{2}} cdot sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x ) A ( cdot frac{5}{72} pi^{2} ) в. ( frac{13}{144} pi^{2} ) c. ( frac{7}{72} pi^{text {? }} ) D. ( frac{1}{12} pi^{2} ) | 12 |
| 460 | ( int frac{x^{4}+1}{x^{6}+1} d x= ) A ( cdot tan ^{-1} x-tan ^{-1} x^{3}+c ) B. ( tan ^{-1} x-frac{1}{3} tan ^{-1}left(x^{3}right)+c ) c. ( tan ^{-1} x+tan ^{-1}left(x^{3}right)+c ) D. ( tan ^{-1} x+frac{1}{3} tan ^{-1}left(x^{3}right)+c ) | 12 |
| 461 | 13. dx is 12004 2 (sin x + cos x) The value of I = 1+ sin 2x (a) 3 (6) 1 (c) 2 0 (d) o | 12 |
| 462 | ( f(x)=frac{4}{pi} sin left(frac{pi}{2} xright)+B ) and ( int_{1}^{0} f(x) d x=frac{4}{pi} int sin left(frac{pi}{2} xright)+B d x, ) Find ( boldsymbol{B} ) | 12 |
| 463 | If ( boldsymbol{I}=int_{0}^{1 / sqrt{3}} frac{boldsymbol{d} boldsymbol{x}}{left(1+boldsymbol{x}^{2}right) sqrt{1-boldsymbol{x}^{2}}} ) then ( boldsymbol{I} ) is equal to ( mathbf{A} cdot pi / 2 ) B. ( pi / 2 sqrt{2} ) c. ( pi / 4 sqrt{2} ) D . ( pi / 4 ) | 12 |
| 464 | 13. Let g(x) = f(t)dt , where fis such that *<f(t) <1, for t e[0,1] and 0 s f(t)55, for t e[1,2]. Then g(2) satisfies the inequality (2000) (b) 05g(2)<2 (0) 3<g(2). (d) 2<g(2)<4 | 12 |
| 465 | The minimum value of the function ( f(x) ) ( =int_{0}^{x} frac{d theta}{cos theta}+int_{x}^{pi / 2} frac{d theta}{sin theta} ) where ( x inleft[0, frac{pi}{2}right], ) is A ( .2 ln (sqrt{2}+1) ) в. ( ln (2 sqrt{2}+2) ) c. ( ln (sqrt{3}+2) ) D. ( ln (sqrt{2}+3) ) | 12 |
| 466 | a noi VC 30. Let f’be a non-negative function de O and f(0) = 0, then [0, 1]. IT JV1-(SO)? dt = sodi, osxsi, (3) < Lands(3) » (200 0 [])–and S(:) < | 12 |
| 467 | ( intleft(x^{2}-5 x+7right) d x ) | 12 |
| 468 | Evaluate the following integral as limit of sums: ( int_{0}^{5}(x+1) d x ) | 12 |
| 469 | Evaluate: ( int_{0}^{pi / 2} frac{8 sin theta+4 cos theta}{sin theta+cos theta} ) | 12 |
| 470 | – 17. Iffand g are continuous function on [0, a] satisfying f(x)=f(a-x) and g(x) + g(a-x)=2, а then show that (x)g(x)dx dx (1989- 4 Marks | 12 |
| 471 | Integrate ( int frac{1}{x^{1 / 2}+x^{1 / 3}} d x ) | 12 |
| 472 | Evaluate the given integral. ( int(x+1) e^{x} log left(x e^{x}right) d x ) | 12 |
| 473 | ( int sec x d x ) | 12 |
| 474 | 8. Evaluate the following Fxsin- dx (1984 – 2 Marks) 0 V1-x 121 | 12 |
| 475 | ( intleft{frac{(log x-1)}{1+(log x)^{2}}right}^{2} d x ) is equals to? A. ( frac{log x}{(log x)^{2}+1}+C ) в. ( frac{x}{x^{2}+1}+C ) c. ( frac{x e^{x}}{1+x^{2}}+C ) D. ( frac{x}{(log x)^{2}+1}+C ) | 12 |
| 476 | ( int x sqrt{frac{a^{2}-x^{2}}{a^{2}+x^{2}}} d x= ) A ( cdot frac{1}{2} a^{2} cos ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{4}+x^{4}}+c ) B ( cdot frac{1}{2} sin ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{4}+x^{4}}+c ) c. ( frac{1}{2} a^{2} sin ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{4}-x^{4}}+c ) D. ( frac{1}{2} cos ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{4}+x^{4}}+c ) | 12 |
| 477 | ( int frac{3 x^{2}}{1+x^{6}} d x ) | 12 |
| 478 | Solve: ( int_{0}^{pi} sqrt{1-sin x} d x ) | 12 |
| 479 | The value of ( left(int x cdot e^{-x} d xright) ) is, | 12 |
| 480 | 2z dz Evaluate: 32+1 | 12 |
| 481 | Solve: ( int frac{(x-1)}{(x+1)(x-2)} d x ) | 12 |
| 482 | Evaluate the following integral ( int frac{a}{b+c e^{x}} d x ) | 12 |
| 483 | The value of the integral ( int_{-1}^{1}left{frac{x^{2013}}{e^{|x|}left(x^{2}+cos xright)}+frac{1}{e^{|x|}}right} d x ) is equal to A. 0 B. ( 1-e^{-1} ) ( c cdot 2 e^{-1} ) D. ( 2left(1-e^{-1}right) ) | 12 |
| 484 | Let ( mathbf{S}_{mathbf{n}}=sum_{mathbf{k}=1}^{mathbf{n}} frac{mathbf{n}}{mathbf{n}^{2}+mathbf{k} mathbf{n}+mathbf{k}^{2}} ) and ( mathbf{T}_{mathbf{n}}= ) ( sum_{mathbf{k}=0}^{mathbf{n}-1} frac{mathbf{n}}{mathbf{n}^{2}+mathbf{k} mathbf{n}+mathbf{k}^{2}} ) for ( mathbf{n}=mathbf{1}, mathbf{2}, mathbf{3}, dots ) Then, This question has multiple correct options A ( cdot mathrm{S}_{mathrm{n}}frac{pi}{3 sqrt{3}} ) ( mathrm{c} cdot mathrm{T}_{mathrm{n}}frac{pi}{3 sqrt{3}} ) | 12 |
| 485 | ( boldsymbol{n} stackrel{boldsymbol{L} t}{rightarrow} inftyleft[frac{boldsymbol{1}^{3}}{boldsymbol{n}^{4}+boldsymbol{1}^{4}}+frac{boldsymbol{2}^{boldsymbol{3}}}{boldsymbol{n}^{4}+mathbf{2}^{4}}+ldots+right. ) ( left.frac{mathbf{1}}{boldsymbol{2} boldsymbol{n}}right]= ) A. ( frac{1}{4} log 4 ) B. ( frac{1}{2} log 2 ) c. ( frac{1}{4} log 3 ) D. ( frac{1}{4} log 2 ) | 12 |
| 486 | Show that ( int_{0}^{pi / 2} f(sin 2 x) sin x d x= ) ( sqrt{2} int_{0}^{pi / 4} f(cos 2 x) cos x d x ) | 12 |
| 487 | Integrate with respect to ( x ) ( int frac{1}{x^{6}left(1+x^{-5}right)^{frac{1}{5}}} d x ) | 12 |
| 488 | The value of ( lim _{n rightarrow infty} Sigma_{1}^{n} cos left(frac{pi}{2}+frac{pi i}{2 n}right) frac{pi}{2 n}=? ) A ( cdot int_{frac{pi}{2}}^{pi} cos x ) B. ( int_{frac{pi}{2}}^{pi} cos x ) ( ^{mathrm{c}} cdot int_{frac{pi}{2}}^{2 pi} cos x ) D. ( int_{frac{pi}{2}}^{5 pi} cos x ) | 12 |
| 489 | The value of ( int_{a}^{b} f(x) d x ) | 12 |
| 490 | ( lim _{n rightarrow infty} sum_{r=1}^{4 n} frac{1}{n+r} ) ( mathbf{A} cdot log _{e} 5 ) B. 0 ( mathbf{c} cdot log _{e} 4 ) D. none of these | 12 |
| 491 | Solve ( int_{0}^{frac{pi}{2}} sqrt{sin phi} cos ^{5} phi d phi ) A ( cdot frac{64}{231} ) в. ( frac{24}{231} ) c. ( frac{54}{231} ) D. None of these | 12 |
| 492 | ff ( frac{mathbf{3 x}+mathbf{4}}{(x+1)^{2}(x-1)}=frac{A}{x-1}+frac{B}{x+1}+ ) ( frac{C}{(x+1)^{2}}, ) then ( C= ) A. ( -frac{1}{2} ) B. ( -frac{1}{4} ) ( c cdot-frac{7}{4} ) D. ( -frac{1}{4} ) | 12 |
| 493 | Integrate: ( intleft(a^{x}+x^{a}+a^{a}right) d x ) A. ( -frac{a^{x}}{ln a}-frac{x^{a+1}}{a+1}+a^{a} x+c ) B. ( a^{x}+frac{x^{a+1}}{a+1}+a^{a} x+c ) c. ( frac{a^{x}}{ln a}+x^{a+1}+a^{a} x+c ) D. ( frac{a^{x}}{ln a}+frac{x^{a+1}}{a+1}+a^{a} x+c ) | 12 |
| 494 | If ( int frac{e^{x}left(2-x^{2}right)}{(1-x) sqrt{1-x^{2}}} d x= ) ( mu e^{x}left(frac{1+x}{1-x}right)^{lambda}+C, ) then ( 2(lambda+mu) ) is equal to ( A ) B. ( c cdot 2 ) D. 3 | 12 |
| 495 | If ( 0<alpha<pi / 2 ) then the value of ( int_{0}^{alpha} frac{boldsymbol{d} boldsymbol{x}}{1-cos boldsymbol{x} cos boldsymbol{alpha}} ) is ( A cdot pi / alpha ) B . ( pi / 2 sin alpha ) c. ( pi / 2 cos alpha ) D. ( pi / 2 alpha ) | 12 |
| 496 | ( int_{0}^{1} frac{2 x}{sqrt{1-x^{4}}} d x ) is equal to? ( A ) в. ( frac{pi}{2} ) c. ( 2 pi ) D. | 12 |
| 497 | Integrate ( int x log 2 x d x ) | 12 |
| 498 | Solve: ( int frac{boldsymbol{d x}}{13+3 cos x+4 sin x} ) | 12 |
| 499 | The value of ( int_{-pi}^{pi} frac{cos ^{2} x}{1+a^{x}} d x, a>0, ) is A ( cdot frac{pi}{2} ) в. ( a pi ) c. ( 2 pi ) D. | 12 |
| 500 | ( lim _{mathbf{n} rightarrow infty} sum_{mathbf{r}=mathbf{1}}^{mathbf{n}} frac{mathbf{1}}{mathbf{n}} mathbf{e}^{mathbf{r} / mathbf{n}} mathbf{i} mathbf{s} ) ( mathbf{A} cdot mathbf{e} ) B. e -1 ( c cdot 1-e ) D. e +1 | 12 |
| 501 | The value of the integral ( int_{0}^{overline{2}} sin ^{5} x d x ) is A ( cdot frac{4}{15} ) B. ( frac{8}{5} ) c. ( frac{8}{15} ) D. ( frac{4}{5} ) | 12 |
| 502 | ( int frac{boldsymbol{x}}{boldsymbol{x}^{4}+boldsymbol{x}^{2}+1} boldsymbol{d} boldsymbol{x} ) ( ( ^{mathbf{B}} cdot=frac{1}{sqrt{3}} tan ^{-1}left(frac{2 x^{2}+1}{sqrt{3}}right)+C ) ( =frac{1}{sqrt{3}} sin ^{-1}left(frac{2 x^{2}+1}{sqrt{3}}right)+C ) C ( =frac{1}{sqrt{2}} tan ^{-1}left(frac{x^{2}+1}{sqrt{2}}right)+C ) ( =frac{1}{sqrt{2}} sin ^{-1}left(frac{x^{2}+1}{sqrt{2}}right)+C ) | 12 |
| 503 | Evaluate ( int_{0}^{3} frac{x}{sqrt{x^{2}+16}} d x ) | 12 |
| 504 | The value of ( int frac{10^{x / 2}}{sqrt{10^{-x}-10^{x}}} d x ) is A. ( frac{1}{log _{2} 10} sin ^{-1}left(10^{x}right)+c ) В. ( 2 sqrt{10^{-x}+10^{x}}+c ) c. ( frac{1}{log _{e} 10} sin h^{-1}left(10^{x}right)+c ) D. ( frac{-1}{log _{e} 10} sin h^{-1}left(10^{x}right)+c ) | 12 |
| 505 | Solve ( int x^{2} cos x d x ) | 12 |
| 506 | Evaluate ( int frac{x^{3}+4 x^{2}-7 x+5}{x+2} d x ) | 12 |
| 507 | If the primitive of ( frac{x^{5}+x^{4}-8}{x^{3}-4 x} ) is ( frac{x^{3}}{3}+ ) ( frac{boldsymbol{x}^{2}}{2}+boldsymbol{A} boldsymbol{x}+|log boldsymbol{f}(boldsymbol{x})|+C ) then This question has multiple correct options ( mathbf{A} cdot A=1 ) ( mathbf{B} cdot A=4 ) ( mathbf{C} cdot f(x)=x^{2}(x-2)^{5}(x+2)^{-3} ) D. ( f(x)=x^{2}(x-2)^{3}(x+2)^{-2} ) | 12 |
| 508 | If ( int frac{f(x)}{1-x^{3}} d x= ) ( log left|frac{x^{2}+x+1}{x-1}right| frac{A}{948 sqrt{3}} tan ^{-1} frac{2 x+1}{sqrt{3}}+ ) ( C ) then ( A=_{-}- ) where ( f(x) ) is a polynomial of second degree in ( x ) such that ( f(0)=f(1)= ) ( mathbf{3} f(mathbf{2})=mathbf{3} ) | 12 |
| 509 | Evaluate the following definite integral: ( int_{e}^{e^{2}}left{frac{1}{log x}-frac{1}{(log x)^{2}}right} d x ) | 12 |
| 510 | ( int sec ^{8 / 9} x operatorname{cosec}^{10 / 9} x d x ) is equal to A. ( -(cot x)^{1 / 9}+c ) B. ( 9(tan x)^{1 / 9}+c ) ( mathbf{c} cdot-9(cot x)^{1 / 9}+c ) D. ( -frac{1}{9}(cot x)^{1 / 9}+c ) | 12 |
| 511 | ( int frac{1}{left(x^{6}-1right)} d x ) A ( cdot 1 / 2left(frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-frac{1}{sqrt{3}} arctan frac{2 x+1}{sqrt{3}}-frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}-right. ) ( left.quad frac{1}{sqrt{3}} arctan frac{2 x-1}{sqrt{3}}right)+mathrm{k} ) B ( cdot ) [ begin{array}{l}text { C } mid / 2left(frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}-frac{1}{sqrt{3}} arctan frac{2 x-1}{sqrt{3}}-frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-right. \ left.frac{1}{sqrt{3}} arctan frac{2 x+1}{sqrt{3}}right)+mathrm{k}end{array} ] C ( cdot 1 / 2left(frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-frac{1}{sqrt{3}} operatorname{arccot} frac{2 x+1}{sqrt{3}}+frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}+right. ) ( left.frac{1}{sqrt{3}} operatorname{arccot} frac{2 x-1}{sqrt{3}}right)+mathrm{k} ) D ( cdot 1 / 2left(frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-frac{1}{sqrt{3}} operatorname{arccot} frac{2 x+1}{sqrt{3}}-frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}-right. ) ( left.frac{1}{sqrt{3}} operatorname{arccot} frac{2 x-1}{sqrt{3}}right)+mathrm{k} ) | 12 |
| 512 | Integrate the function ( frac{cos x}{sqrt{1+sin x}} ) | 12 |
| 513 | If ( boldsymbol{I}=int_{2}^{3} frac{2 boldsymbol{x}^{5}+boldsymbol{x}^{4}-boldsymbol{2} boldsymbol{x}^{3}+boldsymbol{2} boldsymbol{x}^{2}+mathbf{1}}{left(boldsymbol{x}^{2}+mathbf{1}right)left(boldsymbol{x}^{4}-mathbf{1}right)} boldsymbol{d} boldsymbol{x} ) then Iequals A ( cdot frac{1}{2} log 6+frac{1}{10} ) B ( cdot frac{1}{2} log 6-frac{1}{10} ) c. ( frac{1}{2} log 3-frac{1}{10} ) D. ( frac{1}{2} log 2+frac{1}{10} ) | 12 |
| 514 | Evaluate ( int_{0}^{pi / 4} sec ^{2} x d x ) | 12 |
| 515 | If ( I_{n}=int x^{n} e^{a x} d x, ) then ( I_{n}-frac{x^{n} e^{a x}}{a}= ) A. ( frac{n}{a} I_{n-2} ) B. ( -frac{n}{a} I_{n-2} ) c. ( frac{n}{a} I_{n-1} ) D. ( -frac{n}{a} I_{n-1} ) | 12 |
| 516 | Solve:( int frac{x}{sqrt{4-x^{2}}} d x ) | 12 |
| 517 | Evaluate: ( int_{0}^{frac{pi}{2}} log left(frac{4+3 sin x}{4+3 cos x}right) d x ) | 12 |
| 518 | Evaluate: ( int frac{x^{3}-x^{2}+x-1}{x-1} d x ) | 12 |
| 519 | Find : ( int log x d x ) | 12 |
| 520 | Find ( int frac{sqrt{boldsymbol{x}}}{sqrt{boldsymbol{a}^{3}-boldsymbol{x}^{3}}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 521 | Solve ( : int_{0}^{1} sqrt{9-4 x^{2}} d x ) | 12 |
| 522 | ( int frac{sin 2 x}{1+cos ^{4} x} d x ) is equal to ( mathbf{A} cdot cos ^{-1}left(cos ^{2} xright)+c ) B. ( sin ^{-1}left(cos ^{2} xright)+c ) C ( cdot cot ^{-1}left(cos ^{2} xright)+c ) D. None of these | 12 |
| 523 | ( int tan ^{2} x d x ) A ( cdot tan x-x+c ) B. ( tan x+c ) c. ( tan x-x ) D. None of the above | 12 |
| 524 | Evaluate ( int_{0}^{pi / 2} frac{cos ^{2} x}{1+3 sin ^{2} x} d x ) | 12 |
| 525 | Evaluate: ( int frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)} d x ) | 12 |
| 526 | ( frac{x^{2}+3 x+5}{(x+1)(x+2)(x+3)}= ) ( frac{boldsymbol{A}}{(boldsymbol{x}+mathbf{1})}+frac{boldsymbol{B}}{(boldsymbol{x}+mathbf{1})(boldsymbol{x}+mathbf{2})}+ ) ( frac{c}{(x+1)(x+2)(x+3)} ) then ascending order of ( A, B, C ) is ( A cdot B, A, C ) B. A, B, C c. ( c, A, B ) D. B, C, A | 12 |
| 527 | Assertion ( : int frac{1}{sqrt{x^{2}+2 x+10}} d x= ) ( sinh ^{-1} frac{x+1}{3}+c ) Reason : ( operatorname{If} boldsymbol{a}>mathbf{0}, boldsymbol{b}^{2}-mathbf{4} boldsymbol{a} boldsymbol{c}<mathbf{0}, ) then ( int frac{d x}{sqrt{a x^{2}+b x+c}}= ) ( frac{1}{sqrt{a}} sinh ^{-1}left(frac{2 a x+b}{sqrt{4 a c-b^{2}}}right)+k ) A. Both A and R are true and R is the correct explanation of B. Both A and R are true but R is not correct explanation of ( c . ) A is true but ( R ) is false D. A is false but R is true | 12 |
| 528 | The value of ( int_{0}^{pi / 4} frac{sin ^{frac{1}{2}} x}{cos ^{frac{5}{2}} x} d x ) ( mathbf{A} cdot mathbf{0} ) в. c. D. | 12 |
| 529 | Integrate the rational function ( frac{x}{(x-1)^{2}(x+2)} ) | 12 |
| 530 | 74 tan” x dx then lim n[In +In+2) equals n->00 in +In+2] equals [2002] (b) 1 (d) zero (C) | 12 |
| 531 | 12. The integral sin? x cos2x (sinºx+cos’x sin? x +sinx cos²x+cos® x)2 “* zdx is equal to [JEE M 2018) (a) 3(1+tanºx)+C (b) 1+cotx+C -tc © 1+ cotx 3(1+tanºx)+C (where C is a constant of integration) | 12 |
| 532 | The value of ( int(x-1) e^{-x} d x ) is equal to This question has multiple correct options ( mathbf{A} cdot-x e^{x}+C ) B . ( x e^{x}+C ) c. ( -x e^{-x}+C ) D. ( x e^{-x}+C ) | 12 |
| 533 | Evaluate : ( int frac{d x}{a+b e^{c x}} ) | 12 |
| 534 | Maximum value of ( g(x) ) in ( x in[0,7] ) is. ( A cdot 3 ) B. ( 9 / 2 ) ( c .3 / 2 ) ( D ) | 12 |
| 535 | The value of ( int frac{d t}{t^{2}+2 x t+1}left(x^{2}>1right) ) is… ( ^{mathrm{A}} cdot frac{1}{2 sqrt{left(x^{2}-1right)}} log frac{t+x-sqrt{x^{2}-1}}{t+x+sqrt{left(x^{2}+1right)}} ) B. ( frac{1}{2 sqrt{left(x^{2}-1right)}} log frac{t+x-sqrt{x^{2}-1}}{t+x+sqrt{left(x^{2}-1right)}}+c ) c. ( frac{1}{2 sqrt{left(x^{2}+1right)}} log frac{t+x-sqrt{x^{2}+1}}{t+x+sqrt{left(x^{2}+1right)}} ) D. ( frac{1}{2 sqrt{left(x^{2}+1right)}} log frac{t+x-sqrt{x^{2}+1}}{t+x+sqrt{left(x^{2}-1right)}}+c ) | 12 |
| 536 | ( int_{-pi / 2}^{pi / 2} log left(frac{2-sin theta}{2+sin theta}right) d theta=? ) ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. – | 12 |
| 537 | The value of ( left(int_{0}^{pi / 6} sec ^{2} x d xright)^{2} ) is: | 12 |
| 538 | Integrate the following function with respect to ( x: ) ( int frac{1}{left(x^{2}-1right)} d x ) | 12 |
| 539 | If ( int f(x) d x=2{f(x)}^{3}+c, ) and ( f(x) neq 0 ) then ( f(x) ) is A ( cdot frac{x}{2} ) B . ( x^{3} ) c. ( frac{1}{sqrt{x}} ) D. ( sqrt{frac{x}{3}} ) | 12 |
| 540 | Prove that ( boldsymbol{I}=int frac{boldsymbol{t}+mathbf{1}}{left(-boldsymbol{t}^{2}+boldsymbol{t}+mathbf{3}right)} ) | 12 |
| 541 | ( int tan ^{-1} sqrt{frac{1-cos 2 x}{1+cos 2 x}} d x, ) where ( 0< ) ( x<frac{pi}{2} ) is equal to A. ( 2 x^{2}+C ) B. ( x^{2}+C ) c. ( frac{x^{2}}{2}+C ) D. ( frac{x^{3}}{3}+C ) | 12 |
| 542 | Integrate ( int x . sin 2 x d x ) | 12 |
| 543 | Solve ( int_{pi / 2}^{3 pi / 2}[2 sin x] d x ) | 12 |
| 544 | The value of ( int_{-1}^{1} max {2-x, 2,1+x} d x ) is? ( mathbf{A} cdot mathbf{4} ) B. ( frac{9}{2} ) ( c cdot 2 ) D. none of these | 12 |
| 545 | Evaluate ( int_{1}^{3}(2 x+3) d x ) | 12 |
| 546 | ( int frac{boldsymbol{x}^{3 / 2}+boldsymbol{x}^{5 / 2}+boldsymbol{x}}{boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}^{1 / 2}+boldsymbol{x}+1} boldsymbol{d} boldsymbol{x} ) equals A. ( frac{2}{3} x^{3 / 2}+x-tan ^{-1} x^{1 / 4}+C ) B. ( frac{2}{3} x^{5 / 2}-x+tan ^{-1} x^{1 / 2}+C ) C. ( frac{2}{3} x^{1 / 2}-x-tan ^{-1} x^{1 / 4}+C ) D. ( frac{2}{3} x^{3 / 2}-x+tan ^{-1} x^{1 / 4}+C ) | 12 |
| 547 | Integrate ( int_{a}^{b} e^{x} d x ) | 12 |
| 548 | The value of ( lim _{n rightarrow infty} Sigma_{i=1}^{n} log left(2+frac{5 i}{n}right) frac{5}{n} ) is equal to? A ( cdot int_{2}^{5} ln (x) d x ) B. ( int_{3}^{7} ln (x) d x ) c. ( int_{2}^{7} ln (x) d x ) D ( cdot int_{2}^{7} ln left(x^{2}right) d x ) | 12 |
| 549 | ( int_{mathbf{A}}^{boldsymbol{A}} cdot frac{mathbf{1}}{boldsymbol{x}^{2}} sqrt{frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}} boldsymbol{d} boldsymbol{x}= ) B. ( cos ^{-1} frac{1}{|x|}-frac{sqrt{x^{2}-1}}{x}+c ) C ( cdot sin ^{-2} frac{1}{|x|}-frac{sqrt{x^{2}-1}}{x}+c ) D. ( sin ^{-1} frac{1}{|x|}+frac{sqrt{x^{2}-1}}{x}+c ) | 12 |
| 550 | The value of ( int_{0}^{pi} sin ^{50} x cos ^{49} x d x ) is A . в. c. ( frac{pi}{2} ) D. | 12 |
| 551 | Assertion ( (A): ) If ( frac{5 x+1}{(x+2)(x-1)}= ) ( frac{boldsymbol{A}}{(boldsymbol{x}+mathbf{2})}+frac{boldsymbol{B}}{(boldsymbol{x}-mathbf{1})} ) and ( sin boldsymbol{theta}=(boldsymbol{A}+boldsymbol{B}) ) then ( sin theta ) does not exist Reason ( (mathrm{R}): sin theta in[-1,1] ) A. Both A and R are true and R is correct explanation of A B. Both A and R are true and R is not correct explanation of A c. A is true and R is false D. A is false and R is true | 12 |
| 552 | Evaluate: ( int_{frac{pi}{2}}^{pi} frac{1-sin x}{1-cos x} d x ) | 12 |
| 553 | If ( frac{pi}{4}<alpha<frac{pi}{2}, ) value of ( int_{-pi / 2}^{pi / 2} frac{sin 2 x}{sqrt{1+sin 2 alpha sin x}} ) is A ( cdot-frac{4}{3} tan alpha sec alpha ) B. ( -frac{4}{3} cot alpha operatorname{cosec} alpha ) c. ( -frac{4}{3} tan alpha operatorname{cosec} alpha ) D. ( -frac{4}{3} cot alpha sec alpha ) | 12 |
| 554 | Integrate the following ( int frac{x^{2}}{(x sin x+cos x)^{2}} d x ) | 12 |
| 555 | Solve: ( int frac{4 x+6}{2 x^{2}+5 x+3} d x ) | 12 |
| 556 | 1. 101 J sin x | dx is (a) 20 (6) 8 [2002 18 O (c) 10 (d) 14 | 12 |
| 557 | Integrate ( int frac{1}{sqrt{4-x^{2}}} d x ) | 12 |
| 558 | ( int_{0}^{frac{pi}{2}} frac{200 sin x+100 cos x}{sin x+cos x} d x= ) A ( .50 pi ) B . ( 25 pi ) ( c .75 pi ) D. ( 150 pi ) | 12 |
| 559 | Solve ( int_{0}^{pi / 2} frac{x sin x cos x}{cos ^{4} x+sin ^{4} x} d x ) | 12 |
| 560 | ( int_{0}^{1} frac{d x}{x+sqrt{x}}= ) ( A cdot log 2 ) B. 2 log 2 c. ( 3 log 3 ) D. ( frac{1}{2} log 2 ) | 12 |
| 561 | ( int frac{sin 2 x}{sin ^{4} x+cos ^{4} x} d x ) is equal to A ( cdot cot ^{-1}left(tan ^{2} xright)+C ) B. ( tan ^{-1}left(tan ^{2} xright)+C ) C. ( cot ^{-1}left(cot ^{2} xright)+C ) D. ( tan ^{-1}left(cot ^{2} xright)+C ) | 12 |
| 562 | ( int frac{x^{2}left(x sec ^{2} x+tan xright)}{(x tan x+1)^{2}} d x ) ( mathbf{A} cdot x^{2}left[-frac{1}{x tan x+1}right]+2 log (x sin x+cos x)+C ) B ( cdot x^{2}left[frac{1}{x tan x+1}right]+2 log (sin x+x cos x)+C ) ( mathbf{c} cdot xleft[-frac{1}{x tan x+1}right]+2 log (sin x+x cos x)+C ) D ( cdot xleft[frac{1}{x tan x+1}right]+2 log (x sin x+cos x)+C ) | 12 |
| 563 | ( int_{0}^{frac{pi}{2}} frac{sqrt{sin x}}{sqrt{sin x}+sqrt{cos x}} d x ) is equal to ( mathbf{A} cdot mathbf{0} ) в. ( -pi ) c. ( frac{3 pi}{2} ) D. ( frac{pi}{2} ) E. ( frac{pi}{4} ) | 12 |
| 564 | Find ( int frac{x^{4}+1}{xleft(x^{2}+1right)^{2}} ) | 12 |
| 565 | ( int_{0}^{16} frac{d x}{sqrt{x+9}-sqrt{x}}= ) A . 10 B. 12 ( c cdot 14 ) D. 16 | 12 |
| 566 | Solve ( int frac{e^{x}(x-1)}{(x+1)^{3}} d x ) A ( cdot frac{-e^{x}}{(x+1)^{2}}+c ) в. ( frac{e^{x}}{(x+1)^{2}}+c ) c. ( frac{e^{x}}{(x+1)^{3}}+c ) D. ( frac{-e^{x}}{(x+1)^{3}}+c ) | 12 |
| 567 | sec²x The integral ( seca dx equals (for some arbitrary (sec x + tan x)2 constant K) (2012) m (sec x + tan x) 2 (sec x + tan x)2 i lle (600x = tansey) + K 0 l }(eex = tan } * (0) — 1 (1+3(secx + tan x)}+K – 1 1 (1+2(secx + tan x)}} + K (sec x + tan x) 2 (sec x + tan x)2 | 12 |
| 568 | 22. Find det | 12 |
| 569 | The value of the integral ( int_{-a}^{a} frac{x e^{x^{2}}}{1+x^{2}} d x ) is A ( cdot e^{a^{2}}^{2} ) B. ( mathbf{c} cdot e^{-a^{2}}^{2} ) D. ( a ) | 12 |
| 570 | If ( phi(x)=f(x)+x f^{1}(x) ) then ( int phi(x) d x ) is equal to A. ( (x+1) f(x)+k ) В. ( (x-1) f(x)+k ) c. ( x f(x)+k ) D. None of these | 12 |
| 571 | The value of the integral ( int_{-1 / 2}^{1 / 2} cos x cdot log left(frac{1+x}{1-x}right) d x ) A. 0 B. ( c cdot-frac{1}{2} ) D. | 12 |
| 572 | ( int_{-5}^{5} log left(frac{130-x^{3}}{130+x^{3}}right) d x ) is equal to ( mathbf{A} cdot log frac{57}{5} ) B. ( 2 int_{-5}^{5} log left(frac{130-x^{3}}{130+x^{3}}right) d x ) c. 0 D. – | 12 |
| 573 | Using integration, find the area of the triangle ( P Q R, ) whose vertices are at ( boldsymbol{P}(mathbf{2}, mathbf{5}), boldsymbol{Q}(mathbf{4}, mathbf{7}) ) and ( boldsymbol{R}(mathbf{6}, mathbf{2}) ) | 12 |
| 574 | ( int e^{x} sec x(1+tan x) d x ) A ( cdot e^{x} cos x+C ) B . ( e^{x} sec x+C ) ( mathbf{c} cdot e^{x} sin x+C ) D. ( e^{x} tan x+C ) | 12 |
| 575 | Evaluate ( int e^{x}left(log x+frac{1}{x^{2}}right) d x ) A ( cdot e^{x} log x+c ) B. ( e^{x}left(log x-frac{1}{x}right)+c ) c. ( e^{x}left(log x+frac{1}{x}right)+c ) D. ( frac{e^{x}}{x^{2}}+c ) | 12 |
| 576 | Evaluate ( : int e^{sin ^{-1} x}left(frac{ln x}{sqrt{1-x^{2}}}+frac{1}{x}right) d x ) | 12 |
| 577 | Evaluate the given definite integrals as limit of sums: ( int_{-1}^{1} e^{x} d x ) | 12 |
| 578 | ( int frac{e^{x}(1+x)}{cos ^{2}left(x e^{x}right)} d x ) | 12 |
| 579 | Evaluate the following integrals: ( int sec ^{4} 2 x d x ) | 12 |
| 580 | The value of ( int_{-8}^{8}left(sin ^{93} x+x^{295}right) d x ) A . B. – c. 0 D. | 12 |
| 581 | Find the integral ( int frac{d x}{sqrt{9 x-4 x^{2}}} ) | 12 |
| 582 | ( int frac{cos x+sin x}{cos x-sin x} d x ) ( mathbf{A} cdot log sin (pi / 4+x) ) ( mathbf{B} cdot log sec (pi / 4+x) ) ( mathbf{C} cdot log cos (pi / 4+x) ) D ( cdot log sec (pi / 4-x) ) | 12 |
| 583 | Integrate the rational function ( frac{boldsymbol{x}}{(boldsymbol{x}+mathbf{1})(boldsymbol{x}+mathbf{2})} ) | 12 |
| 584 | Integrate :- ( int log log x+frac{1}{(log x)^{2}} d x ) | 12 |
| 585 | Integrate the following w.r.t. ( x ) ( frac{1}{2 x+3} ) | 12 |
| 586 | ( int frac{1}{7 x+6} d x ) | 12 |
| 587 | If ( frac{1-x+6 x^{2}}{x-x^{3}}=frac{A}{x}+frac{B}{1-x}+frac{C}{1+x} ) then ( mathbf{A}= ) ( A ) B . 2 ( c .3 ) D. 4 | 12 |
| 588 | ( boldsymbol{I}=int sqrt[3]{boldsymbol{x}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 589 | The value of ( int frac{e^{x}}{x}(x log x+1) d x ) is equal to A ( cdot frac{e^{x}}{x}+C ) B . ( x e^{x} log |x|+C ) c. ( e^{x} log |x|+C ) D・ ( xleft(e^{x}+log |x|right)+C ) E ( cdot x e^{x}+log |x|+C ) | 12 |
| 590 | Evaluate ( int_{0}^{pi / 2} cos x d x ) | 12 |
| 591 | Evaluate ( int frac{boldsymbol{d x}}{sqrt{mathbf{2 a x}-boldsymbol{x}^{2}}} ) | 12 |
| 592 | The value of ( int_{0}^{pi / 2} frac{cos 3 x+1}{2 cos x-1} d x ) is equal to A . 2 B. c. ( frac{1}{2} ) D. | 12 |
| 593 | Evaluate the given integral: ( int_{0}^{1} x^{4} d x ) | 12 |
| 594 | ( int x^{9} d x ) | 12 |
| 595 | Area bounded by ( mathbf{y}={mathbf{x}},{.} ) is fractional part of function and ( mathbf{x}=pm mathbf{1} ) is in sq. units ( mathbf{A} cdot mathbf{1} ) B. 2 ( c cdot 3 ) D. 4 | 12 |
| 596 | ntegrate the function ( frac{mathbf{5} boldsymbol{x}+mathbf{3}}{sqrt{boldsymbol{x}^{2}+mathbf{4} boldsymbol{x}+mathbf{1 0}}} ) | 12 |
| 597 | Solve: ( int sin ^{3} x cdot cos ^{2} x d x ) | 12 |
| 598 | Taking constant of integration as zero, find ( f(1) ) ( int frac{x e^{x}}{(x+1)^{2}} d x ) | 12 |
| 599 | ( int sec ^{2} x cdot operatorname{cosec}^{2} x d x= ) ( mathbf{A} cdot tan x-cot x+c ) B. ( tan x+cot x+c ) c. ( -tan x+cot x+c ) ( D cdot sec x tan x+c ) | 12 |
| 600 | Antiderivative of ( frac{sin ^{2} x}{1+sin ^{2} x} ) with respect to x is? A ( cdot x-frac{sqrt{2}}{2} arctan (sqrt{2} tan x)+c ) B. ( x-frac{1}{sqrt{2}} arctan left(frac{tan x}{sqrt{2}}right)+c ) c. ( x-sqrt{2} a r c tan (sqrt{2} tan x)+c ) D. ( x-sqrt{2} arctan left(frac{tan x}{sqrt{2}}right)+c ) | 12 |
| 601 | Evaluate: ( intleft(frac{x cos x+sin x}{x sin x}right) d x ) | 12 |
| 602 | Evaluate the definite integral, ( int_{-1}^{1} frac{left(x^{332}+x^{998}+4 x^{1668} cdot sin x^{691}right)}{1+x^{666}} d x ) A ( cdot frac{2}{333} ) в. ( frac{1}{333} ) c. ( frac{4}{33} ) D. ( frac{5}{333} ) | 12 |
| 603 | State whether True=1 or False=0 ( int frac{x^{2}}{left(x^{2}+1right)left(x^{2}+4right)} d x=frac{-1}{3} tan ^{-1} x+ ) ( frac{2}{3} tan ^{-1}left(frac{x}{2}right)+C ) | 12 |
| 604 | Evaluate: ( int tan ^{-1} x d x ) | 12 |
| 605 | Evaluate : ( int frac{x}{(x-1)^{2}(x+2)} d x ) | 12 |
| 606 | The value of ( int frac{1}{sqrt{sin ^{3} x cos ^{5} x}} d x ) is A ( cdot frac{-2}{sqrt{tan x}}+frac{2}{3}(tan x)^{3 / 2}+C ) в. ( frac{2}{sqrt{tan x}}-frac{2}{3}(tan x)^{3 / 2}+C ) c. ( frac{-2}{sqrt{tan x}}+frac{2}{3}(tan x)^{1 / 2}+C ) D. None of these | 12 |
| 607 | ( int frac{cos ^{2} x}{1+tan x} d x ) A ( cdot frac{1}{4} ln (cos -sin x)+frac{x}{2}+frac{1}{8}(sin 2 x-cos 2 x) ) B. ( frac{1}{4} ln (cos +sin x)+frac{x}{2}+frac{1}{8}(sin 2 x+cos 2 x) ) C ( frac{1}{4} ln (cos +sin x)+frac{x}{2}+frac{1}{8}(sin 2 x-cos 2 x) ) ( frac{1}{4} ln (cos -sin x)+frac{x}{2}+frac{1}{8}(sin 2 x+cos 2 x) ) | 12 |
| 608 | Evaluate: ( int_{0}^{1} frac{1}{sqrt{1+x}-sqrt{x}} d x ) | 12 |
| 609 | Integrate: ( int x sqrt{x^{2}+2} d x ) | 12 |
| 610 | Integrate: ( int_{0}^{pi} frac{d x}{5+3 cos x} ) | 12 |
| 611 | Find the antiderivative of the function ( left(sin frac{x}{2}+cos frac{x}{2}right)^{2} ) | 12 |
| 612 | ( frac{boldsymbol{x}^{2}+mathbf{5}}{left(boldsymbol{x}^{2}+mathbf{2}right)^{2}}=frac{mathbf{1}}{boldsymbol{x}^{2}+mathbf{2}}+frac{boldsymbol{k}}{left(boldsymbol{x}^{2}+mathbf{2}right)^{2}} Rightarrow ) ( boldsymbol{k}= ) ( A ) B. ( c cdot 3 ) D. 5 | 12 |
| 613 | Integrate the following function: ( e^{x}left(frac{1+sin x}{1+cos x}right) ) | 12 |
| 614 | Let ( f(x)=sqrt{5 x-1} ) and let ( c ) be the number that satisfies the Mean value theorem for ( f ) on the interval [1,10] Find the value of ( c ) A . 2.25 B. 3.25 5 c. 4.25 D. None of the above | 12 |
| 615 | Evaluate: ( int frac{x^{4}+1}{1+x^{6}} d x ) A ( cdot tan ^{-1}(x)-tan ^{-1}left(x^{3}right)+c ) B cdot ( tan ^{-1}(x)-frac{1}{3} tan ^{-1}left(x^{3}right)+c ) c. ( tan ^{-1}(x)+tan ^{-1}left(x^{3}right)+c ) D. ( tan ^{-1}(x)+frac{1}{3} tan ^{-1}left(x^{3}right)+c ) | 12 |
| 616 | Evaluate: ( int x cos ^{3} x d x ) | 12 |
| 617 | Evaluate: ( int x^{-9} d x ) | 12 |
| 618 | Evaluate ( int frac{(x+sqrt{1+x^{2}})^{15}}{sqrt{1+x^{2}}} d x ) A. ( frac{(x+sqrt{1+x^{2}})^{14}}{14}+C ) B. ( frac{(x+sqrt{1+x^{2}})^{15}}{15}+C ) c. ( frac{(x+sqrt{1+x^{2}})^{16}}{16}+C ) D. ( frac{(x+sqrt{1+x^{2}})^{17}}{17}+C ) | 12 |
| 619 | 1 x+ 8. The integral [|1+x — ex dx is equal to (JEE M 2014] х x +- (a) (x+1)(x + (b) -xe +c (C) (x-1)e***+C x+ (d), xe x +c | 12 |
| 620 | Let ( S_{n}=sum_{k=1}^{n} frac{n}{n^{2}+k n+k^{2}} ) and ( T_{n}= ) ( sum_{k=0}^{n-1} frac{n}{n^{2}+k n+k^{2}}, ) for ( n=1,2,3, dots ) Then, This question has multiple correct options A ( cdot S_{n}frac{pi}{3 sqrt{3}} ) c. ( T_{n}frac{pi}{3 sqrt{3}} ) | 12 |
| 621 | Solve: ( int frac{sin ^{3} x+cos ^{3} x}{sin ^{2} x cos ^{2} x} d x ) | 12 |
| 622 | ( operatorname{Let} boldsymbol{I}= ) ( left.int_{3 n pi}^{left(n+frac{1}{n}right) 3 pi} frac{mathbf{4} boldsymbol{x} boldsymbol{d} boldsymbol{x}}{left[left(boldsymbol{a}^{2}+boldsymbol{b}^{2}right)+left(boldsymbol{a}^{2}-boldsymbol{b}^{2}right) cos frac{2 n boldsymbol{x}}{3}right.}right] ) (where ( mathbf{a}, mathbf{b}>mathbf{0}) ) prove that ( boldsymbol{I}=frac{mathbf{9}left(mathbf{2} boldsymbol{n}^{2}+mathbf{1}right) boldsymbol{pi}}{boldsymbol{n}^{2}} frac{boldsymbol{a}^{2}-boldsymbol{b}^{2}}{boldsymbol{a}^{3} boldsymbol{b}^{3}} ) | 12 |
| 623 | If ( int_{a}^{b} frac{f(x)}{f(a)+f(a+b-x)} d x=10, ) then This question has multiple correct options A. ( b=22, a=2 ) В. ( b=15, a=-5 ) c. ( b=10, a=-10 ) D. ( b=10, a=-2 ) | 12 |
| 624 | ( int(x+5)^{3} d x . ) Integrate this using fundamental properties of indefinite integral. | 12 |
| 625 | ( int(e x)^{x}(2+log x) d x=ldots .+c, x in ) ( boldsymbol{R}^{+}-{mathbf{1}} ) ( mathbf{A} cdot x^{x} ) B. ( (e x)^{x} ) ( mathbf{c} cdot e^{x} ) D ( cdot(1+log x)(e x)^{x} ) | 12 |
| 626 | Find ( : int frac{sin 2 x}{left(sin ^{2} x+1right)left(sin ^{2} x+3right)} d x ) | 12 |
| 627 | Evaluate : ( int log x d x ) | 12 |
| 628 | ( int_{frac{1}{sqrt{3}}}^{0} frac{boldsymbol{d} boldsymbol{x}}{left(mathbf{2} boldsymbol{x}^{2}+mathbf{1}right) sqrt{boldsymbol{x}^{2}+mathbf{1}}} ) A ( cdot-tan ^{-1} frac{1}{2} ) B. ( tan ^{-1} 1 ) c. ( -tan ^{-1} frac{1}{3} ) D. ( tan ^{-1} frac{1}{sqrt{2}} ) | 12 |
| 629 | ( int sqrt{1-sin x} d x= ) A. ( 2 sqrt{1+sin x}+C ) B . ( 2 sqrt{1-sin x}+C ) c. ( 2 sqrt{1-2 sin x}+C ) D. ( 2 sqrt{1-sin 2 x}+C ) | 12 |
| 630 | Solve : ( int frac{cos x-sin x)}{(1+sin 2 x)} d x ) | 12 |
| 631 | Suppose we know that ( f(x) ) is continuous and differentiable on the interval ( [-7,0], ) that ( f(-7)=-3 ) and that ( f^{prime}(x) leq 2 . ) What is the largest possible value for ( boldsymbol{f}(mathbf{0}) ? ) A . 12 B. 22 c. 11 D . 24 | 12 |
| 632 | If ( frac{(x+1)^{2}}{xleft(x^{2}+1right)}=frac{A}{x}+frac{B x+C}{x^{2}+1}, ) then ( cos ^{-1}left(frac{A}{C}right)= ) A. ( frac{pi}{6} ) в. c. D. | 12 |
| 633 | If ( f(x) ) is a function satisfying ( fleft(frac{1}{x}right)+ ) ( x^{2} f(x)=0 ) for all non-zero ( x, ) then ( int_{sin theta}^{operatorname{cosec} theta} f(x) d x ) equals to: A ( cdot sin theta+operatorname{cosec} theta ) B. ( sin ^{2} theta ) ( mathrm{c} cdot operatorname{cosce}^{2} theta ) D. None of these | 12 |
| 634 | x + – 2r + x 26. Evalute + 1 dx. (1993 – 5 Marks | 12 |
| 635 | Let ( boldsymbol{f}(boldsymbol{x}), boldsymbol{g}(boldsymbol{x}) ) and ( boldsymbol{h}(boldsymbol{x}) ) be continuous function on ( [0, a] ) such that ( f(x)= ) ( boldsymbol{f}(boldsymbol{a}-boldsymbol{x}), boldsymbol{g}(boldsymbol{x})=-boldsymbol{g}(boldsymbol{a}-boldsymbol{x}), boldsymbol{3} boldsymbol{h}(boldsymbol{x})- ) ( 4 h(a-x)=5 ) then ( int_{0}^{a} f(x) g(x) h(x) d x ) is equal to A . 1 B. ( c ) D. – | 12 |
| 636 | ( int_{0}^{1} x(1-x)^{4} d x=frac{1}{C}, ) then ( C=? ) | 12 |
| 637 | ( int_{-1}^{1} x|x| d x ) is equal to A ( cdot frac{2}{3} ) B. ( -frac{2}{3} ) c. 0 D. None of these | 12 |
| 638 | Find ( F(x) ) from the ( operatorname{given} F^{prime}(x) ) ( F^{prime}(x)=2 sin 5 x+3 cos (x / 2) ) which is zero for ( boldsymbol{x}=boldsymbol{pi} / mathbf{3} ) | 12 |
| 639 | The value of ( int_{-pi}^{pi} frac{cos ^{2} x}{1+a^{x}} d x, a>0 ) is ( mathbf{A} cdot pi / 2 ) в. ( a pi ) c. ( pi ) D. ( 2 pi ) | 12 |
| 640 | Solve : ( int frac{d x}{sqrt{1-x^{2}}}=sin ^{-1} x+c ) | 12 |
| 641 | ( int_{0}^{1} frac{sqrt{x}}{1+x} d x= ) A . ( 2-pi / 2 ) B. ( 1-pi / 2 ) c. ( pi / 2 ) D. ( 2+pi / 2 ) | 12 |
| 642 | Evaluate the integral ( int_{0}^{a} sqrt{a^{2}-x^{2}} d x ) A ( cdot frac{a^{2}}{4} ) B ( cdot pi a^{2} ) c. ( frac{pi a^{2}}{2} ) D. ( frac{pi a^{2}}{4} ) | 12 |
| 643 | ( int frac{cos x}{cos (x-a)} d x ) | 12 |
| 644 | =1-sin x, then f 2. sin x (c) 3 | 12 |
| 645 | Evaluate the definite integral ( int_{frac{pi}{6}}^{frac{pi}{3}} frac{sin x+cos x}{sqrt{sin 2 x}} d x ) | 12 |
| 646 | The value of ( int_{-pi}^{pi}left(1-x^{2}right) sin x cos ^{2} x d x ) is A. 0 в. ( _{pi}-frac{pi^{3}}{3} ) c. ( 2 pi-pi^{3} ) D. ( frac{7}{2}-2 pi^{3} ) | 12 |
| 647 | Integrate: ( int frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{x}^{5}left(1+boldsymbol{x}^{-4}right)} ) | 12 |
| 648 | ( int e^{x} frac{x-1}{(x+1)^{3}} d x ) is equal to A ( cdot frac{e^{x}}{x+1}+C ) в. ( frac{e^{x}}{(x+1)^{2}}+C ) c. ( -frac{e^{x}}{x+1}+C ) D. ( -frac{e^{x}}{(x+1)^{2}}+C ) | 12 |
| 649 | Integrate: ( int sin ^{4} x d x ) | 12 |
| 650 | If ( int_{0}^{k} frac{cos x}{1+sin ^{2} x} d x=frac{pi}{4} ) then ( k=? ) A . в. ( pi / 4 ) c. ( pi / 2 ) D. ( pi / 6 ) | 12 |
| 651 | Integrate: ( frac{3 x^{2}}{x^{6}+1} ) | 12 |
| 652 | Integrate the function ( e^{x}(sin x+ ) ( cos x ) | 12 |
| 653 | 9. The integral (-7 dx, 4 equals : [JEE M 2 X (x + 1 -(x + 1) 4 + c (e) (x + 1) + (%) (x + 1) + c (d) (x++1)4 + c | 12 |
| 654 | What is the value of ( int_{0}^{a} frac{x-a}{x+a} d x ? ) A. ( a+2 a log 2 ) в. ( a-2 a log 2 ) c. ( 2 a log 2-a ) D. ( 2 a log 2 ) | 12 |
| 655 | The value of ( int_{2}^{3} frac{sqrt{x}}{sqrt{5-x}+sqrt{x}} d x ) is A . 1 B. ( c cdot 2 ) D. None of these | 12 |
| 656 | ( operatorname{Let} f(x)=int frac{x^{2} d x}{left(1+x^{2}right)(1+sqrt{1+x^{2}})} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} . ) Then ( boldsymbol{f}(mathbf{1}) ) is A ( cdot log (1+sqrt{2}) ) B ( cdot log (1+sqrt{2})-frac{pi}{4} ) ( c cdot log (1+sqrt{2})+frac{pi}{4} ) D. none of these | 12 |
| 657 | Evaluate the given integral. ( int frac{log (log x)}{x} d x ) | 12 |
| 658 | ( int frac{2 x+sin 2 x}{1+cos 2 x} d x ) ( A cdot x cot x ) B. ( x tan x ) c. ( x^{2} tan x ) D. ( x ) | 12 |
| 659 | The value of ( 3 int_{0}^{pi / 2} sqrt{cos x-cos ^{3} x} d x ) is | 12 |
| 660 | ( int frac{e^{cot ^{-1} x}}{1+x^{2}}left(x^{2}-x+1right) d x ) ( ^{A} cdot frac{e^{cot ^{-1} x}}{1+x^{2}} ) B. ( x cdot e^{cot ^{-1} x} ) ( mathbf{c} cdot e^{cot ^{-1} x} ) D. ( -e^{cot ^{-1} x} ) | 12 |
| 661 | Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) be continuous functions, then the value of ( int_{-frac{pi}{2}}^{frac{pi}{2}}(f(x)+f(-x))(g(x)-g(-x)) d x ) is equal to ( A ) B. ( c cdot 1 ) D. none of these | 12 |
| 662 | The value of the integral ( int_{0}^{pi / 2} log [sin (x)] d x ) is ( A cdot log 2 ) B. ( -log 2 ) c. ( frac{pi}{2} log 2 ) D. ( -frac{pi}{2} log 2 ) | 12 |
| 663 | By using the properties of definite integrals, evaluate the integral ( int_{0}^{frac{pi}{2}} frac{sqrt{sin x}}{sqrt{sin x}+sqrt{cos x}} d x ) | 12 |
| 664 | Write a value of ( int sqrt{x^{2}-9} d x ) | 12 |
| 665 | Evaluate ( intleft(frac{x^{2}+3 x+4}{sqrt{x}}right) d x ) | 12 |
| 666 | Integrate the rational function ( frac{1}{left(e^{x}-1right)} ) | 12 |
| 667 | Evaluate ( : int_{0}^{log _{e} 5} frac{e^{x} sqrt{e^{x}-1}}{e^{x}+3} d x ) | 12 |
| 668 | ( int_{-pi / 2}^{pi / 2} tan x^{3} d x=? ) A . 1 B. ( c cdot 2 ) D. | 12 |
| 669 | ( int 5^{m x} 7^{n x} d x, m, n in N ) is equal to This question has multiple correct options ( mathbf{A} cdot frac{5^{m x}+7^{n x}}{m log 5+n log 7}+K ) B. ( frac{e^{(m log 5+n log 7) x}}{log 5^{m}+log 7^{n}}+K ) C ( cdot frac{(m cdot n) 5^{m x}+7^{n x}}{m log 5+n log 7}+K ) D. None of these | 12 |
| 670 | Integrate the rational function ( frac{1}{xleft(x^{n}+1right)} ) | 12 |
| 671 | Evaluate: ( int sqrt{frac{boldsymbol{a}+boldsymbol{x}}{boldsymbol{a}-boldsymbol{x}}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 672 | By the definition of the definite integral, the value of ( lim _{n rightarrow infty}left(frac{1^{4}}{1^{5}+n^{5}}+frac{2^{4}}{2^{5}+n^{5}}+frac{3^{4}}{3^{5}+n^{5}}+right. ) is ( A cdot log 2 ) B cdot ( frac{1}{5} log 2 ) c. ( frac{1}{4} log 2 ) D. ( frac{1}{3} log 2 ) | 12 |
| 673 | Solve: ( int frac{x^{2}-1}{x^{3} sqrt{2 x^{4}-2 x^{2}+1}} d x ) is equal to A ( cdot frac{sqrt{2 x^{4}-2 x^{2}+1}}{x^{2}}+C ) B. ( frac{sqrt{2 x^{4}-2 x^{2}+1}}{x^{3}}+C ) c. ( frac{sqrt{2 x^{4}-2 x^{2}+1}}{x}+C ) D. ( frac{sqrt{2 x^{4}-2 x^{2}+1}}{2 x^{2}}+C ) | 12 |
| 674 | ( int frac{1}{left[left(1-x^{2}right)left{left(2 sin ^{-1} xright)^{2}-9right}right]^{1 / 2}} d x ) A. ( frac{1}{2} log left[2 sin ^{-1} x+sqrt{left(2 sin ^{-1} xright)^{2}-9}right] ) B. ( log left[2 sin ^{-1} x+sqrt{left(2 sin ^{-1} xright)^{2}-9}right] ) c. ( frac{1}{2} log left[2 sin ^{-1} x+sqrt{left(2 sin ^{-1} xright)^{2}+9}right] ) D. ( frac{1}{2} log left[2 sin ^{-1} x+sqrt{left(2 cos ^{-1} xright)^{2}-9}right] ) | 12 |
| 675 | Solve ( int frac{x^{2}}{left(x^{2}+1right)left(x^{2}+4right)} d x ) | 12 |
| 676 | ( int_{-3 pi / 2}^{-pi / 2}left[(x+pi)^{3}+cos ^{2}(x+3 pi)right] d x ) is equal to ( mathbf{A} cdotleft(pi^{4} / 32right)+(pi / 2) ) B. ( pi / 2 ) ( mathbf{c} cdot(pi / 4)-1 ) ( mathbf{D} cdotleft(pi^{4} / 32right) ) | 12 |
| 677 | ( int_{0}^{frac{pi}{2}} frac{sin ^{3} x}{sin x+cos x} d x ) is equal to? A ( cdot frac{pi}{4}-frac{1}{4} ) B. ( frac{pi}{4}+frac{1}{4} ) c. ( frac{pi}{4}+frac{1}{2} ) D. ( frac{pi}{4}-frac{1}{2} ) | 12 |
| 678 | What is ( int_{-frac{pi}{2}}^{frac{pi}{2}}|sin x| d x ) equal to ( ? ) ( A cdot 2 ) B. 1 ( c . pi ) D. | 12 |
| 679 | Evaluate ( int e^{x}left(frac{1+sin x}{1+cos x}right) ) | 12 |
| 680 | 15. The value of the integral dx is: (2000S) (a) 3/2 (b) 5/2 (C) 3 (d) 5 | 12 |
| 681 | ( int frac{6 x+7}{(x-5)(x-4)} d x ) | 12 |
| 682 | Evaluate: ( int frac{x^{3}-4 x^{2}+6 x+5}{x^{2}-2 x+3} d x ) | 12 |
| 683 | Solve: ( int_{0}^{pi / 4} tan ^{3} x sec x d x ) | 12 |
| 684 | ( boldsymbol{I}=int frac{1}{2 x^{2}+3 x+4} d x ) | 12 |
| 685 | 7. S v1+ cos x dx equals (a) 212 sin+C (C) -2 12cos+c (6) -212 sin+c (d) 272.cos+c | 12 |
| 686 | Evaluate the integral ( int_{1}^{sqrt[7]{2}} frac{1}{xleft(2 x^{7}+1right)} d x ) ( ^{mathrm{A}} cdot log frac{6}{5} ) B. ( 6 log frac{6}{5} ) c. ( frac{1}{7} log _{5}^{6} ) D. ( frac{1}{5} log frac{6}{5} ) | 12 |
| 687 | Evaluate the given integral. ( int sqrt{frac{x}{1-x}} d x ) ( A cdot sin ^{-1} sqrt{x}+C ) ( mathbf{B} cdot sin ^{-1}[sqrt{x}-sqrt{x(1-x)}]+C ) ( mathbf{c} cdot sin ^{-1}[sqrt{x(1-x)}]+C ) D. ( sin ^{-1} sqrt{x}-sqrt{x(1-x)}+C ) | 12 |
| 688 | ( frac{x^{3}}{x^{2}-x+2}=x+k-left[frac{x+2}{x^{2}-x+2}right] Rightarrow ) ( mathbf{k}= ) ( mathbf{A} cdot mathbf{4} ) B . 2 ( c cdot 1 ) ( D ) | 12 |
| 689 | Integrate ( int frac{x^{3}-x^{2}+x-1}{x-1} d x ) | 12 |
| 690 | ntegrate the function ( frac{1}{sqrt{8+3 x-x^{2}}} ) | 12 |
| 691 | Solve : ( int frac{3 x+5}{sqrt{7 x+9}} d x ) | 12 |
| 692 | ( int frac{cos x}{sin ^{2} x cdot(sin x+cos x)} d x ) is equal to ( mathbf{A} cdot log left|frac{1+tan x}{tan x}right|-cot x+C ) ( ^{mathbf{B}} cdot log left|frac{1+tan x}{tan x}right|+C ) ( ^{mathbf{C}} log left|frac{1+tan x}{tan x}right|-tan x+C ) D ( cdot log left|frac{1+tan x}{tan x}right|+cot x+C ) | 12 |
| 693 | ( int x e^{2 x}(1+x) d x ) equal to A ( cdot frac{x e^{x}}{2}+c ) B ( cdot frac{left(e^{x}right)^{2}}{2} r ) c. ( frac{(1+x)^{2}}{2}+c ) D. ( frac{left(x e^{x}right)^{2}}{2} ) | 12 |
| 694 | Solve ( int_{0}^{pi / 2}(2 log sin x-log sin 2 x) d x ) A ( cdot frac{pi}{2} log 2 ) B. ( -frac{pi}{2} log 2 ) c. ( frac{pi}{4} log 2 ) D. None | 12 |
| 695 | ( int_{log 1 / 2}^{log 2} sin left(frac{e^{x}-1}{e^{x}+1}right) d x ) is equal to A ( cdot cos frac{1}{3} ) B. c. ( 2 cos 2 ) D. none of these | 12 |
| 696 | What is ( int_{0}^{1} frac{tan ^{-1} x}{1+x^{2}} d x ) equal to ( ? ) A ( cdot frac{pi}{4} ) в. c. ( frac{pi^{2}}{8} ) D. ( frac{pi^{2}}{32} ) | 12 |
| 697 | Integrate ( int frac{log x}{x^{2}} d x ) | 12 |
| 698 | Evaluate ( : int_{-x}^{x}(cos a x-sin b x)^{2} d x ) | 12 |
| 699 | 27. j *+ 3×2 + 3x+3+ (x + 1)cos(x + 1)} der is equal to 3x + 3x +3+ (x + 1) cos(x+1)} dx is equal to | 12 |
| 700 | ( int sin ^{-1} d x ) | 12 |
| 701 | Evaluate the following definite integral: ( int_{0}^{4}left(4 x-x^{2}right) d x ) | 12 |
| 702 | Evaluate ( int_{0}^{1} cot ^{-1}left(1-x+x^{2}right) d x ) | 12 |
| 703 | Solve ( sqrt{frac{x^{2}-a^{2}}{x}} d x ) | 12 |
| 704 | The mean value of the function ( f(x)= ) ( frac{2}{e^{x}+1} ) on the interval [0,2] is A ( cdot log frac{2}{e^{2}+1} ) B. ( 1+log frac{2}{e^{2}+1} ) c. ( _{2+log _{e^{2}+1}} frac{2}{e} ) D. ( 2+log left(e^{2}+1right) ) | 12 |
| 705 | Solve: ( int_{0}^{1} x+x^{2} d x ) | 12 |
| 706 | ( int frac{2^{x}}{sqrt{1-4^{x}}} d x=K sin ^{-1}left(2^{x}right)+C, ) then the value of ( K ) is equal to A ( . e n 2 ) B. ( frac{1}{2} ell 2 ) ( c cdot frac{1}{2} ) D. ( frac{1}{ell n^{2}} ) | 12 |
| 707 | ( boldsymbol{I}=int frac{1}{sqrt{2 x^{2}+3 x+8}} d x ) | 12 |
| 708 | Find ( int frac{d x}{xleft(x^{3}+1right)^{2}} ) | 12 |
| 709 | Find the integrals of the functions. i) ( sin ^{3} x cos ^{3} x ) ii) ( sin x sin 2 x sin 3 x ) iii) ( sin 4 x sin 8 x ) iv ( frac{1-cos x}{1+cos x} ) v) ( frac{cos x}{1+cos x} ) | 12 |
| 710 | ( int_{0}^{pi / 2} frac{1}{a^{2} sin ^{2} x+b^{2} cos ^{2} x} d x ) | 12 |
| 711 | A positive integer ( n leq 5 ), such that ( int_{0}^{1} e^{2 x-1}(x-1)^{n} d x=frac{1}{4}left(frac{7}{e}-eright) ) | 12 |
| 712 | ( int_{0}^{pi / 2} x sin x cos x d x ) | 12 |
| 713 | f ( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{x}^{3} sqrt{boldsymbol{x}^{2}-1}}, ) then I equals A ( cdot frac{1}{2}left(frac{sqrt{x^{2}-1}}{x^{3}}+tan ^{-1} sqrt{x^{2}-1}right)+C ) B. ( frac{1}{2}left(frac{sqrt{x^{2}-1}}{x^{2}}+x tan ^{-1} sqrt{x^{2}-1}right)+C ) ( ^{c} cdot frac{1}{2}left(frac{sqrt{x^{2}-1}}{x}+tan ^{-1} sqrt{x^{2}-1}right)+C ) D ( -frac{1}{2}left(frac{sqrt{x^{2}-1}}{x^{2}}+tan ^{-1} sqrt{x^{2}-1}right)+C ) | 12 |
| 714 | ( intleft[frac{1+sin (log x)}{1+cos (log x)}right] d x= ) A. ( frac{x}{1+cos (log x)}+c ) в. ( quad x tan frac{log x}{2}+c ) c. ( quad-x cot frac{log x}{2}+c ) D. ( frac{x}{1+sin (log x)}+c ) | 12 |
| 715 | 38. Ifg(x) = (cos 4t dt , then g(x + 1) equals CO (b) g(x)+g(1) et 8 (T) (c) g(x)-g(1) (d) g(x).g (T) | 12 |
| 716 | Evaluate the integral ( int_{0}^{2}left(x^{2}+2 x+right. ) 1) ( d x ) | 12 |
| 717 | Solve ( int(2 x+3)^{2} d x ) | 12 |
| 718 | Evaluate the following definite integral: ( int_{-1}^{1} frac{1}{x^{2}+2 x+5} d x ) | 12 |
| 719 | Integrate ( frac{x^{2}+5 x+5}{x^{2}+3 x+2} ) with respect to ( x ) | 12 |
| 720 | ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{(1+boldsymbol{x}) sqrt{left(boldsymbol{2}+boldsymbol{x}-boldsymbol{x}^{2}right)}}=frac{mathbf{1}}{boldsymbol{k}} sqrt{mathbf{2}} . ) Finc the value of ( k ) | 12 |
| 721 | Integrate ( int frac{3 x-1}{(x-1)(x-2)(x-3)} d x ) | 12 |
| 722 | Solve : ( int frac{d x}{x^{2}+8 x+20} ) | 12 |
| 723 | Evaluate: ( int frac{x}{sqrt{left(1-x^{2}right)} cos ^{2} sqrt{left(1-x^{2}right)}} d x ) A ( cdot tan sqrt{1-x^{2}} ) ( mathbf{B} cdot-tan sqrt{1-x^{2}} ) c. ( -tan left(1-x^{2}right) ) D. ( -sec ^{2} sqrt{1-x^{2}} ) | 12 |
| 724 | Solve ( int_{0}^{pi / 2} frac{x sin x cos x}{cos ^{4} x+sin ^{4} x} d x ) | 12 |
| 725 | Evaluate the following integral: ( int frac{x^{2}-1}{x^{2}+4} d x ) | 12 |
| 726 | Using (iiii) above the best upper bound of ( int_{0}^{1} sqrt{1+x^{4}} d x ) A . 1.2 B. ( sqrt{1.22} ) c. ( sqrt{1.2} ) D. ( sqrt{1.4} ) | 12 |
| 727 | ( int_{0}^{pi / 2} frac{sin 8 x log (cot x) d x}{cos 2 x} ) | 12 |
| 728 | If ( int frac{1}{5+4 cos 2 theta} d theta=A tan ^{-1}(B tan theta)+c ) then ( (A, B)= ) | 12 |
| 729 | The value of ( int_{-pi / 2}^{pi / 2} sqrt{frac{1}{2}(1-cos 2 x)} d x ) is ( A cdot 0 ) B. 2 ( c cdot frac{1}{2} ) D. None of these. | 12 |
| 730 | ( f f(x)=left|begin{array}{ccc}x & cos x & e^{|x|} \ sin x & x^{2} & sec x \ tan x & 1 & 2end{array}right| ) then the value of ( int_{-pi / 2}^{pi / 2} f(x) d x ) is equal to | 12 |
| 731 | ( int_{0}^{infty}left(cot ^{-1} xright)^{2} d x=frac{pi}{k} log 2 . ) Find the value of ( k ) | 12 |
| 732 | The value of ( int cos (log x) d x ) is A ( cdot frac{1}{2}[sin (log x)+cos (log x)]+C ) B. ( frac{x}{2}[sin (log x)+cos (log x)]+C ) c. ( frac{x}{2}[sin (log x)-cos (log x)]+C ) D ( cdot frac{1}{2}[sin (log x)-cos (log x)]+C ) | 12 |
| 733 | dr is equal to cos x -sin x ( tezlog col () +C | 12 |
| 734 | Evaluate the following integral as limit of sum: ( int_{0}^{5}(x+1) d x ) | 12 |
| 735 | If ( I=int_{1}^{infty} frac{x^{2}-2}{x^{3} sqrt{x^{2}-1}} d x, ) then ( I ) equals A . -1 B. c. ( pi / 2 ) D. ( pi-sqrt{3} ) | 12 |
| 736 | Evaluate: ( int frac{1}{x+sqrt{x}} d x ) | 12 |
| 737 | Evaluate: ( int_{e}^{e^{2}} frac{d x}{x log x} ) | 12 |
| 738 | If ( f(x) ) is an even function, and ( n in N ) then ( int_{-pi}^{pi} boldsymbol{f}(boldsymbol{x}) sin boldsymbol{n} boldsymbol{x} boldsymbol{d} boldsymbol{x}= ) A . B ( cdot 2 int_{0}^{pi} f(x) sin n x d x ) c. ( 4 int_{0}^{frac{pi}{2}} f(x) sin n x d x ) D. ( int_{0}^{pi} f(x) sin x d x ) | 12 |
| 739 | ( lim _{n rightarrow infty} sum_{r=1}^{n} frac{1}{sqrt{n r}} ) is equal to ( A cdot 2 ) B. ( c cdot 0 ) D. none of these | 12 |
| 740 | Evaluate ( int_{0}^{1}left(2 x^{2}+x+1right) d x ) | 12 |
| 741 | Evaluate: ( int frac{5 x-2}{1+2 x+3 x} d x ) | 12 |
| 742 | Evaluate: ( int_{0}^{pi / 2} sin ^{3} x cdot cos ^{3} x d x ) A ( cdot frac{1}{12} ) в. ( frac{pi}{24} ) c. ( frac{pi}{12} ) D. ( frac{1}{24} ) | 12 |
| 743 | Evaluate the integral ( int_{2}^{3}left(x^{2}+2 x+right. ) 5) ( d x ) | 12 |
| 744 | ( int frac{x}{sqrt{left(4-x^{4}right)}} d x ) A ( cdot sin ^{-1}left(frac{1}{2} x^{2}right) ) B ( cdot frac{1}{2} sin ^{-1}left(x^{2}right) ) C ( cdot frac{1}{2} sin ^{-1}left(frac{1}{2} x^{2}right) ) D. ( frac{1}{2} cos ^{-1}left(frac{1}{2} x^{2}right) ) | 12 |
| 745 | ( int e^{x}(sin x+2 cos x) sin x d x ) is equal to A ( cdot e^{x} cos x+C ) B. ( e^{x} sin x+C ) ( mathbf{c} cdot e^{x} sin ^{2} x+C ) D. ( e^{x} sin 2 x+C ) E ( cdot e^{x}(cos x+sin x)+C ) | 12 |
| 746 | If ( int frac{x^{4}+1}{x^{6}+1} d x=tan ^{-1}(f(x)) ) ( frac{2}{3} tan ^{-1}(g(x))+C, ) then A. Both ( f(x) & g(x) ) are odd functions B. ( g(x) ) is monotonic function c. none of these D. None | 12 |
| 747 | ( boldsymbol{I}=int log [boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}] boldsymbol{d} boldsymbol{x} ) ( mathbf{A} cdot x log [x+sqrt{x^{2}+a^{2}}]-sqrt{x^{2}+a^{2}} ) B ( cdot x log [x+sqrt{x^{2}+a^{2}}]+x^{2}+a^{2} ) C ( x log [x+sqrt{x^{2}+a^{2}}]+sqrt{x^{2}+a^{2}} ) D. ( x log [x+sqrt{x^{2}+a^{2}}]-x^{2}+a^{2} ) | 12 |
| 748 | If ( int frac{d x}{x^{2}+a x+1}=f(g(x))+c, ) then This question has multiple correct options A ( cdot f(x) ) is inverse trigonometric function for ( |a|>2 ) B . ( f(x) ) is logarithmic function for ( |a|2 ) D ( cdot f(x) ) is logarithmic function for ( |a|>2 ) | 12 |
| 749 | Calculate the following integral ( int_{0}^{3}left[3^{1-x}+left(frac{1}{3}right)^{2 x-1}right] d x ) | 12 |
| 750 | State whether the given statement is True or False ( int_{0}^{2} e^{x^{2}} d x ) can be represented as ( 2 lim _{n rightarrow infty} frac{1}{n}left[e^{0}+e^{frac{4}{n^{2}}}+e^{frac{16}{n^{2}}}+ldots ldots+e^{frac{2(n-1)^{2}}{n^{2}}}right] ) A. True B. False | 12 |
| 751 | ( int frac{2 sin x}{(3+sin 2 x)} d x ) is equal to ( mathbf{A} cdot frac{1}{2} ln left|frac{2+sin x-cos x}{2-sin x+cos x}right| ) ( frac{1}{sqrt{2}} tan ^{-1} xleft(frac{sin x+cos x}{sqrt{2}}right)+c ) B. ( frac{1}{2} ln left|frac{2+sin x-cos x}{2-sin x+cos x}right| ) ( frac{1}{2 sqrt{2}} tan ^{-1} xleft(frac{sin x+cos x}{sqrt{2}}right)+c ) C ( frac{1}{4} ln left|frac{2+sin x-cos x}{2-sin x+cos x}right| ) ( frac{1}{sqrt{2}} tan ^{-1} xleft(frac{sin x+cos x}{sqrt{2}}right)+c ) D. none of these | 12 |
| 752 | f ( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{left(boldsymbol{e}^{boldsymbol{x}}+mathbf{2}right)^{3}}, ) then I equals A ( cdot frac{1}{8} x-frac{1}{8} log left(e^{x}+2right)+frac{e^{x}+3}{4left(e^{x}+2right)^{2}}+C ) B. ( frac{1}{8} x+frac{1}{8} log left(e^{x}+2right)+frac{e^{x}}{4left(e^{x}+2right)^{2}}+C ) c. ( frac{1}{8} x+frac{1}{8} log left(e^{x}+2right)+frac{e^{x}}{left(e^{x}+2right)^{2}}+C ) D. none of these | 12 |
| 753 | If ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{f}(boldsymbol{x})=boldsymbol{g}(boldsymbol{x}) ) for ( boldsymbol{a} leq boldsymbol{x} leq boldsymbol{b}, ) then ( int_{a}^{b} f(x) g(x) d x ) equals to: A. ( f(2)-f(1) ) В. ( g(2)-g(1) ) c. ( frac{[f(b)]^{2}-[f(a)]^{2}}{2} ) D. ( frac{[g(b)]^{2}-[g(a)]^{2}}{2} ) | 12 |
| 754 | ( int_{1}^{2} frac{1-x}{1+x} d x ) equals ( left(frac{1}{2}right) log left(frac{3}{2}right)-1 ) B ( cdot 2 log left(frac{3}{2}right)-1 ) ( ^{mathbf{C}} cdot log left(frac{3}{2}right)-1 ) D. None of these | 12 |
| 755 | If ( I_{1}=int_{1}^{2} x(sqrt{x}+sqrt{3-x}) d x ) and ( boldsymbol{I}_{2}=int_{1}^{2}(sqrt{boldsymbol{x}}+sqrt{mathbf{3 – x}}) d boldsymbol{x}, ) then ( frac{boldsymbol{I}_{1}}{boldsymbol{I}_{2}}= ) ( A ) ( overline{2} ) B. ( frac{3}{2} ) ( c cdot 2 ) ( D ) | 12 |
| 756 | ( int frac{e^{x}}{left(e^{x}+2right)left(e^{x}-1right)} d x= ) A ( cdot frac{1}{3} log left|frac{e^{x}-1}{e^{x}+2}right|+c ) в ( cdot frac{1}{3} log left|frac{e^{x}+1}{e^{x}-1}right|+c ) c. ( frac{1}{3} log left|frac{e^{x}+1}{e^{x}+2}right|+c ) D. ( -frac{1}{3} log left|frac{e^{x}+1}{e^{x}+2}right|+c ) | 12 |
| 757 | Evaluate ( int_{0}^{2} x sqrt{x+2} d x ) | 12 |
| 758 | If ( int frac{d x}{left(x^{2}+1right)left(x^{2}+4right)}=k tan ^{-1} x+ ) ( l tan ^{-1} frac{x}{2}+C, ) then A ( cdot quad k=frac{1}{3} ) B ( cdot l=frac{2}{3} ) c. ( quad k=-frac{1}{3} ) D ( cdot l=-frac{1}{6} ) | 12 |
| 759 | ( int frac{log (x+1)-log x}{x(x+1)} d x ) equals | 12 |
| 760 | Integrate: ( int tan ^{3} x d x ) | 12 |
| 761 | Integrate: ( int frac{x^{4}-x^{3}+8 x-8}{x^{2}-2 x+4} d x ) | 12 |
| 762 | Using definite integration, find area of the triangle with vertices at ( A(1,1), B(3,3) A(1,1), B(3,3) ) | 12 |
| 763 | Resolve ( frac{1}{boldsymbol{x}^{4}+1} ) into partial fractions. A ( frac{(x+sqrt{2})}{4 sqrt{2}left(x^{2}+x sqrt{2}+1right)}-frac{(x-sqrt{2})}{4 sqrt{2}left(x^{2}-x sqrt{2}+1right)} ) B. ( frac{(x+sqrt{2})}{sqrt{2}left(x^{2}+x sqrt{2}+1right)}-frac{(x-sqrt{2})}{sqrt{2}left(x^{2}-x sqrt{2}+1right)} ) ( ^{mathbf{C}}-frac{(x+sqrt{2})}{2 sqrt{2}left(x^{2}+x sqrt{2}+1right)}+frac{(x-sqrt{2})}{2 sqrt{2}left(x^{2}-x sqrt{2}+1right)} ) D. ( frac{(x+sqrt{2})}{2 sqrt{2}left(x^{2}+x sqrt{2}+1right)}-frac{(x-sqrt{2})}{2 sqrt{2}left(x^{2}-x sqrt{2}+1right)} ) | 12 |
| 764 | Find the value of ( int frac{boldsymbol{d}left(boldsymbol{x}^{2}+mathbf{1}right)}{sqrt{left(boldsymbol{x}^{2}+mathbf{2}right)}} ) | 12 |
| 765 | ( int frac{e^{x}-1}{e^{x}+1} d x ) A ( cdot log left(e^{x}+1right)-log e^{x} ) B ( cdot 2 log left(e^{x}-1right)-log e^{x} ) C ( cdot 2 log left(e^{x}+1right)-log e^{x} ) D ( cdot 2 log left(e^{x}+1right)+log e^{x} ) | 12 |
| 766 | Evaluate the integral ( int_{0}^{1} cos ^{-1} x d x ) A . B. – c. ( frac{pi}{2} ) D. | 12 |
| 767 | ( int_{-a}^{a} frac{x^{4} d x}{sqrt{a^{2}-x^{2}}}= ) A ( cdot frac{3 pi a^{4}}{8} ) в. ( frac{pi a^{4}}{8} ) c. ( frac{-pi a^{4}}{8} ) D. ( frac{5 pi mathrm{a}^{4}}{8} ) | 12 |
| 768 | 24. tesco is differentiable and [vas()dx =Şe, then s (a) 2/5 (b) -5/2 @ 1 (d) 512->) equals | 12 |
| 769 | Solve: ( int_{0}^{frac{pi}{2}} x^{2} sin x d x ) | 12 |
| 770 | If ( boldsymbol{f} ) satisfies ( |boldsymbol{f}(boldsymbol{u})-boldsymbol{f}(boldsymbol{nu})| leq|boldsymbol{u}-boldsymbol{nu}| ) for ( boldsymbol{u}, boldsymbol{nu} in[boldsymbol{a}, boldsymbol{b}] ) then the maximum value of ( left|int_{a}^{b} f(x) d x-(b-a) f(a)right| ) is? A ( cdot frac{b-a}{2} ) в. ( frac{(b-a)^{2}}{2} ) c. ( (b-a)^{2} ) D. None of these | 12 |
| 771 | Evaluate the given integral. ( int e^{x}(cot x+log sin x) d x ) | 12 |
| 772 | Evaluate : ( int sqrt{frac{1-cos 2 x}{1+cos 2 x}} d x ) | 12 |
| 773 | Solve: ( int frac{log x}{(1+log x)^{2}} d x ) | 12 |
| 774 | ( int(sin x)^{99}(cos x)^{-101} d x=_{-} ldots-C_{ } ) A. ( frac{(tan x)^{100}}{100} ) B. ( frac{(tan x)^{2}}{2} ) c. ( frac{(tan x)^{98}}{98} ) D. ( frac{(tan x)^{97}}{97} ) | 12 |
| 775 | The average value of the pressure varying from 2 to 10 atm if the pressure p and the volume ( v ) are related by ( boldsymbol{p} boldsymbol{v}^{3 / 2}=mathbf{1 6 0} ) is – | 12 |
| 776 | Solve : ( int frac{1-cot x}{1+cot x} d x ) | 12 |
| 777 | Solve: ( int_{0}^{pi / 2} frac{sin x d x}{(sin x+cos x)^{3}} ) | 12 |
| 778 | 6. sin nx If I, = _dx n=0, 1, 2, …, then -(1+*)sin x (20 10 (b) (a) In = In+2 Ce 12m = 0 12m+1 =101 m=1 (d) In = In + 1 m =1 | 12 |
| 779 | ( int_{-pi / 2}^{pi / 2} sqrt{cos ^{2 n-1} x-cos ^{2 n+1} x} d x, ) where | 12 |
| 780 | Find the antiderivative of the function ( left(sin frac{x}{2}+cos frac{x}{2}right)^{2} ) | 12 |
| 781 | If ( sin ^{-1} frac{2 x}{1+x^{2}} ; cos ^{-1} frac{1-x^{2}}{1+x^{2}} ; tan ^{-1} frac{2 x}{1-x^{2}} ) each is equal to ( 2 tan ^{-1} x . ),then show that ( int 2 tan ^{-1} x= ) ( 2left[x tan ^{-1} x-frac{1}{2} log left(1+x^{2}right)right] ) | 12 |
| 782 | 45. Evaluate Jelcosal (2 sin (= cos x) + 3 cos ( cosx)) sin x dx (2005 2 Marks) | 12 |
| 783 | Evaluate the integral ( int_{2}^{3} frac{sqrt{boldsymbol{x}}}{sqrt{mathbf{5}-boldsymbol{x}}+sqrt{boldsymbol{x}}} boldsymbol{d} boldsymbol{x} ) A ( cdot 1 / 2 ) B . ( 3 / 2 ) ( c cdot 5 / 2 ) D. 0 | 12 |
| 784 | U T4 TT/2 18. Show that I f (sin 2x) sin x dx = V2 ) f(cos 2x) cos x de (1990 – 4 Marks | 12 |
| 785 | Evaluate ( int(7 x-2) sqrt{3 x+2} d x ) | 12 |
| 786 | ( int x sec ^{-1} x d x= ) ( frac{2}{k}left[x^{2} sec ^{-1} x-sqrt{x^{2}-1}right] . ) Find the value of ( k ) | 12 |
| 787 | The mean value of 6,9,12 is | 12 |
| 788 | Integrate with respect to ( x ). ( e^{x} sin x ) | 12 |
| 789 | ( intleft(cot ^{n+2} x+cot ^{n} xright) d x= ) A. ( quad frac{-cot ^{n+1} x}{n+1}+c ) B ( cdot frac{-cot ^{n-1} x}{n-1}+C ) c. ( frac{-cot ^{n+3} x}{n+3}+c ) D. ( frac{-cot ^{2 n} x}{2 n}+C ) | 12 |
| 790 | 32. Let f be a real-valued function defined on the interval (-1, 1) such that e*f(x 4 +1 dt, for all xe(-1,1), and let f’l be the inverse function of f. Then (F-1) (2) is equal to (2010) | 12 |
| 791 | Evaluate ( int frac{1}{sqrt{9-25 x^{2}}} d x ) | 12 |
| 792 | Evaluate: ( int_{0}^{pi / 2} frac{sin ^{2} x}{sin x+cos x} d x ) | 12 |
| 793 | The value of ( int frac{boldsymbol{d} boldsymbol{x}}{(boldsymbol{x}-boldsymbol{beta}) sqrt{(boldsymbol{x}-boldsymbol{alpha})(boldsymbol{beta}-boldsymbol{x})}}, ) is equal to ( ^{text {A }} frac{-1}{beta-alpha} sqrt{frac{x-alpha}{beta-x}}+C ) в. ( frac{1}{beta-alpha} sqrt{frac{x-alpha}{beta-x}}+C ) c. ( frac{2}{alpha-beta} sqrt{frac{x-alpha}{beta-x}}+C ) D. none of these | 12 |
| 794 | Solve: ( int frac{d x}{sqrt{x^{2}+2 x+5}} ) A ( cdot ln |sqrt{x^{2}+2 x+5}-x+1|+C ) B ( cdot ln |sqrt{x^{2}+2 x+5}+x+1|+C ) c. ( ln |sqrt{x^{2}+2 x+5}-x|+C ) D. None of these | 12 |
| 795 | ( int(log x)^{2} d x= ) A ( cdot xleft[(log x)^{2}-2 log x+2right]+c ) B . ( xleft[(log x)^{2}+2 log x+2right]+c ) C ( cdotleft[(log x)^{2}-2 log x+2right]+c ) D. ( left[(log x)^{2}+2 log x+2right]+c ) | 12 |
| 796 | Evaluate the integral ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x ) | 12 |
| 797 | The value of ( int_{1 / 2}^{1} frac{d x}{x sqrt{3 x^{2}+2 x-1}} ) is? ( mathbf{A} cdot pi / 2 ) в. ( pi / 3 ) c. ( pi / 6 ) D. ( pi / sqrt{2} ) | 12 |
| 798 | Solve : ( int 2^{x} cdot e^{x} d x ) | 12 |
| 799 | ( operatorname{Let} f(x)=int frac{e^{x}}{x} d x ) and ( int frac{left(e^{x-1}right)(2 x)}{x^{2}-5 x+4} d x=alpha f(x-4)+ ) ( beta f(x-1)+gamma, ) then This question has multiple correct options ( mathbf{A} cdot ln 3 alpha=3 ) B. ( 4+3 beta=ln 3 alpha ) c. ( 3 beta+2=0 ) D. ( ln 3 alpha=3+ln 8 ) | 12 |
| 800 | Evaluate: ( int_{0}^{1}left(8 x^{2}+16right) d x ) | 12 |
| 801 | Integrate ( int_{0}^{pi} frac{e^{cos x}}{e^{cos x}+e^{-cos x}} d x ) ( A cdot frac{pi}{12} ) ( B cdot frac{pi}{3} ) ( mathbf{C} cdot frac{pi}{4} ) D. ( frac{pi}{2} ) | 12 |
| 802 | ( int sqrt{frac{cos x-cos ^{3} x}{1-cos ^{3} x}} d x= ) A. ( frac{2}{3} sin ^{-1}left(cos ^{frac{3}{2}} xright)+c ) B. ( frac{3}{2} sin ^{-1}left(cos ^{frac{3}{2}} xright)+c ) C. ( frac{2}{3} cos ^{-1}left(cos ^{frac{3}{2}} xright)+c ) D. ( frac{3}{2} cos ^{-1}left(cos ^{frac{3}{2}} xright)+c ) | 12 |
| 803 | The value of integral ( int tan ^{-1}left(frac{x^{3}}{1+x^{2}}right)+ ) ( tan ^{-1}left(frac{1+x^{2}}{x^{3}}right) d x ) is equal to ( A ) в. ( -frac{pi}{2}+c ) c. ( frac{pi}{2}+c ) D. ( left(frac{pi}{2}right) x+c ) | 12 |
| 804 | Let ( boldsymbol{f} ) be a positive function. If ( boldsymbol{I}_{mathbf{1}}= ) ( int_{1-k}^{k} x f x(1-x) d x, I_{2}= ) ( int_{1-k}^{k} f x(1-x) d x, ) where ( 2 k-1>0 ) then ( frac{boldsymbol{I}_{1}}{boldsymbol{I}_{2}} ) is A . 2 B. ( k ) ( c cdot frac{1}{2} ) D. | 12 |
| 805 | Find ( int_{0}^{1 / 4 pi} ln (1+tan x) d x ) | 12 |
| 806 | Evaluate the given integral. ( int frac{x^{9}}{left(4 x^{2}+1right)^{6}} d x ) ( ^{mathrm{A}} frac{1}{5 x}left(4+frac{1}{x^{2}}right)^{-5}+C ) в. ( frac{1}{5}left(4+frac{1}{x^{2}}right)^{-5}+C ) ( ^{mathrm{c}} frac{1}{10 x}left(frac{1}{x^{2}}+4right)^{-5}+C ) ( ^{mathrm{D}} frac{1}{10}left(frac{1}{x^{2}}+4right)^{-5}+C ) | 12 |
| 807 | ( int frac{2 x}{sqrt{1-x^{2}-x^{4}}} d x ) | 12 |
| 808 | ( int_{0}^{infty} frac{d x}{[x+sqrt{x^{2}+1}]^{3}} ) is equal to A ( cdot frac{3}{8} ) B. ( frac{1}{8} ) ( c cdot-frac{3}{8} ) D. none of these | 12 |
| 809 | VINOJ 14. For any natural number m, evaluate |(x3m + x2m + x)(2x2m +3xm +6)/m dx ,x>0 | 12 |
| 810 | Evaluate: ( int frac{1}{x^{2}left(x^{4}+1right)^{frac{3}{4}}} d x ; x=0 ) ( ^{mathrm{A}} frac{left(x^{4}-1right)^{frac{1}{4}}}{x}+c ) B. ( -frac{left(x^{4}+1right)^{frac{1}{4}}}{x}+c ) c. ( frac{sqrt{x^{4}+1}}{x}+c ) D. None of these | 12 |
| 811 | Evaluate; ( int_{0}^{pi / 2} log sin 2 x d x ) | 12 |
| 812 | ( int_{0}^{1} sqrt{frac{mathbf{x}}{1-mathbf{x}^{3}}} mathbf{d x}= ) A ( cdot frac{pi}{4} ) в. c. D. | 12 |
| 813 | ( int frac{1-cos x}{cos x(1+cos x)} d x ) A. ( log (sec x-tan x)-2 tan frac{x}{2} ) B. ( log (sec x+tan x)-tan frac{x}{2} ) C ( cdot log (sec x+tan x)+2 tan frac{x}{2} ) D ( cdot log (sec x+tan x)-2 tan frac{x}{2} ) | 12 |
| 814 | ( int frac{t^{2}}{t^{3}+1} d t= ) | 12 |
| 815 | n-1 =1, 2, 3, ……. Then, (2005 ” (a) S. 57 | 6 ins (6) Sa> (0) T. 5 | 12 |
| 816 | Find ( a, b ) in ( int frac{x+2}{left(x^{2}+3 x+3right) sqrt{x+1}} d x= ) ( frac{boldsymbol{a}}{sqrt{boldsymbol{b}}} tan ^{-1}left{frac{boldsymbol{x}}{sqrt{mathbf{3}(boldsymbol{x}+mathbf{1})}}right}+boldsymbol{C} ) This question has multiple correct options ( mathbf{A} cdot a=2 ) B. ( b=3 ) ( mathbf{c} cdot a=3 ) ( mathbf{D} cdot b=2 ) | 12 |
| 817 | Assertion Statement 1 If ( n ) is positive integer then ( int_{0}^{n pi}left|frac{sin x}{x}right| d x geq ) ( frac{2}{pi}left(1+frac{1}{2}+frac{1}{3}+ldots+frac{1}{n}right) ) Reason Statement ( 2 frac{sin x}{x} geq frac{2}{pi} ) on ( (0, pi / 2) ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 818 | Integrate the function ( x log 2 x ) | 12 |
| 819 | Evaluate the integral ( int_{0}^{pi / 2} frac{sqrt{cot x}}{sqrt{tan x}+sqrt{cot x}} d x ) A . ( pi ) B . ( pi / 2 ) c. ( pi / 3 ) D . ( pi / 4 ) | 12 |
| 820 | Find ( : int frac{sin ^{6} x}{cos ^{8} x} d x ) | 12 |
| 821 | 1. Integrate: | 12 |
| 822 | Solve: ( int_{0}^{infty}left(a^{-x}-b^{-x}right) d x ) | 12 |
| 823 | ( int frac{d x}{x^{2}+2 x+2} ) is equal to: ( mathbf{A} cdot sin ^{-1}(x+1)+c ) B. ( sinh ^{-1}(x+1)+c ) c. ( tanh ^{-1}(x+1)+c ) D. ( tan ^{-1}(x+1)+c ) | 12 |
| 824 | ( int e^{x sec x} cdot sec x(1+x tan x) d x= ) A ( cdot e^{x sec x}+c ) B. ( -e^{x sec x}+c ) c. ( frac{1}{e^{x sec x}}+c ) D. ( -e^{x tan x}+c ) | 12 |
| 825 | 9. If y = sin(2x + 3) then ſy dr will be: cos (2x + 3) cos(2x +3). (b) — (c) cos (2x + 3) (d) -2 cos(2x + 3) (a) 2 2 | 12 |
| 826 | The value of ( int e^{x}left[frac{1+sin x}{1+cos x}right] d x ) is A ( cdot frac{1}{2} e^{x} sec frac{x}{2}+C ) B ( cdot e^{x} sec frac{x}{2}+C ) c. ( frac{1}{2} e^{x} tan frac{x}{2}+C ) D. ( e^{x} tan frac{x}{2}+C ) | 12 |
| 827 | If ( boldsymbol{I}=int_{0}^{1} frac{boldsymbol{e}^{t}}{mathbf{1}+boldsymbol{t}} ) dt, then ( boldsymbol{p}= ) ( int_{0}^{1} e^{t} log (1+t) d t= ) ( A ) B. ( 2 I ) c. ( e log 2-I ) D. none | 12 |
| 828 | Solve ( int(2 t-4)^{-4} d t ) | 12 |
| 829 | Evaluate ( int_{0}^{infty} sin x d x ) | 12 |
| 830 | The solution of ( int_{sqrt{2}}^{x} frac{d t}{sqrt{t^{2}-1}}=frac{pi}{12} ) is A . B. 2 ( c cdot 3 ) D. 4 E . 5 | 12 |
| 831 | Illustration 2.43 Evaluate V1+ y2 + 2y dy | 12 |
| 832 | Evaluate the following definite integrals as limit of sums. ( int_{0}^{5}(x+1) d x ) A ( cdot frac{15}{2} ) в. ( frac{25}{2} ) c. ( frac{35}{2} ) D. ( frac{45}{2} ) | 12 |
| 833 | Evaluate ( int_{0}^{2} 3 x+2 d x ) | 12 |
| 834 | ( int sin x log (sec x+tan x) d x=f(x)+ ) ( boldsymbol{x}+boldsymbol{c} ) then ( boldsymbol{f}(boldsymbol{x})= ) A ( cdot cos x log (sec x+tan x)+c ) B. ( sin x log (sec x+tan x)+c ) c. ( -cos x log sec x+tan x)+c ) D. – ( cos x log sec x+c ) | 12 |
| 835 | Integrate w.r.t ( times frac{3 x}{1+2 x^{4}} ) A ( cdot frac{3}{sqrt{2}} tan ^{-1} sqrt{2} x^{2}+c ) B. ( frac{3}{2 sqrt{2}} tan ^{-1} sqrt{2} x^{2}+c ) ( frac{3}{2 sqrt{2}} tan ^{-1} 2 x^{2}+c ) D ( cdot frac{3}{sqrt{2}} tan ^{-1} x^{2}+c ) | 12 |
| 836 | Solve: ( intleft(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}right)^{mathbf{3}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 837 | Evaluate ( int_{0}^{2}left(x^{2}+4right) d x ) | 12 |
| 838 | ( operatorname{Let} boldsymbol{S}_{boldsymbol{n}}=sum_{boldsymbol{k}=1}^{n} frac{boldsymbol{n}}{boldsymbol{n}^{2}+boldsymbol{k} boldsymbol{n}+boldsymbol{k}^{2}}, boldsymbol{T}_{boldsymbol{n}}= ) ( sum_{k=0}^{n-1} frac{n}{n^{2}+k n+k^{2}} ) for ( n=1,2,3 dots ) Then This question has multiple correct options A ( cdot S_{n}frac{pi}{3 sqrt{3}} ) c. ( T_{n}frac{pi}{3 sqrt{3}} ) | 12 |
| 839 | OULD Let f be a real-valued function defined on the interval (0 ) by f(x) = In x + 1+ sint dt. Then which of the following statement(s) is (are) true? (2010) (a) f”(x) exists for all x e(0,00) (b) f'(x) exists for all x € (0,00) and f’ is continuous on (0,00), but not differentiable on (0,00) (c) there exists a > 1 such that f'(x) \ f (x) for all x e(a,0) (d) there exists B> 0 such that | f (x)]+f'(x)|B for all x +(0, ) | 12 |
| 840 | Evaluate the following definite integral: ( int_{0}^{1} frac{2 x+3}{5 x^{2}+1} d x ) | 12 |
| 841 | ( int sin ^{-1} sqrt{frac{x}{a+x}} d x=dots ) A ( cdot(a+x) tan ^{-1} sqrt{x / a}+sqrt{a x}+c ) B. ( (a+x) tan ^{-1} sqrt{x / a}-sqrt{a x}+c ) C. ( (a+x) cot ^{-1} sqrt{x / a}+sqrt{a x}+c ) D. ( (a-x) cot ^{-1} sqrt{x / a}-sqrt{a x}+c ) | 12 |
| 842 | ( int frac{(sin x)^{99}}{(cos x)^{101}} d x=-ldots-ldots+c ) A ( cdot frac{(tan x)^{97}}{97} ) в. ( frac{tan x}{2} ) C ( cdot frac{(tan x)^{100}}{100} ) D. ( frac{(tan x)^{98}}{98} ) | 12 |
| 843 | What is ( int_{0}^{2 pi} sqrt{1+sin frac{x}{2}} d x ) equal to? ( A cdot 8 ) B. 4 ( c cdot 2 ) D. | 12 |
| 844 | 3. Evaluate [(elog x + sin x) cos x dx. Evalua dx | 12 |
| 845 | Evaluate ( int frac{boldsymbol{d} boldsymbol{x}}{sqrt{mathbf{8}+mathbf{3} boldsymbol{x}-boldsymbol{x}^{2}}} ) | 12 |
| 846 | Integrate: ( int frac{1+109 x}{x cdot 109 x} cdot d x ) | 12 |
| 847 | Evaluate the given integral. ( int e^{x} frac{x-4}{(x-2)^{2}} d x ) | 12 |
| 848 | If ( int frac{boldsymbol{d x}}{boldsymbol{a} e^{m x}+boldsymbol{b} e^{-boldsymbol{m} boldsymbol{x}}}= ) ( K tan ^{-1}left(P e^{m x}right)+C, ) then ( K, P= ) ( ^{mathbf{A}} cdot_{K}=frac{1}{sqrt{a b}}, P=sqrt{frac{a}{b}} ) B. ( K=frac{1}{m sqrt{a b}}, P=sqrt{frac{a}{b}} ) ( ^{mathbf{c}} cdot_{K}=m sqrt{a b}, P=sqrt{frac{b}{a}} ) D. ( quad K=frac{1}{m sqrt{a b}}, P=sqrt{frac{b}{a}} ) | 12 |
| 849 | If ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{1}{boldsymbol{x}}+mathbf{3} boldsymbol{x}^{2} ) then ( boldsymbol{y}= ) A. ( ln x+frac{x^{3}}{2}+c ) B. ( ln x+3 x^{3}+c ) c. ( ln x+frac{x^{3}}{3}+c ) D. ( ln x+x^{3}+c ) | 12 |
| 850 | Evaluate ( int_{2}^{3} x^{2}+2 x+5 d x ) | 12 |
| 851 | ( int(3 x-2) sqrt{x^{2}+x+1} d x= ) | 12 |
| 852 | The solution of differential equation ( boldsymbol{x}^{2} boldsymbol{y} boldsymbol{d} boldsymbol{x}-left(boldsymbol{x}^{3}+boldsymbol{y}^{3}right) boldsymbol{d} boldsymbol{y}=mathbf{0} ) is A ( cdot-frac{1}{3} frac{x^{3}}{y^{3}}+log y=C ) B. ( -frac{1 x^{3}}{3 y^{3}}-log y=C ) ( ^{mathbf{c}} cdot frac{x^{3}}{y^{3}}+log y=C ) D. None of these | 12 |
| 853 | ( int frac{1}{left(1+x^{2}right) sqrt{p^{2}+q^{2}left(tan ^{-1} xright)^{2}}} d x ) is equal to A ( cdot frac{1}{p} log left|pleft(tan ^{-1} xright)+sqrt{p^{2}+left(q tan ^{-1} xright)^{2}}right|+C ) B. ( frac{1}{p} log left|left(p cot ^{-1} xright)+sqrt{p^{2}+left(p cot ^{-1} xright)^{2}}right|+C ) c. ( frac{1}{q} log left|qleft(cot ^{-1} xright)+sqrt{p^{2}+left(q cot ^{-1} xright)^{2}}right|+C ) D ( cdot frac{1}{q} log left|qleft(tan ^{-1} xright)+sqrt{p^{2}+left(q tan ^{-1} xright)^{2}}right|+C ) | 12 |
| 854 | ( int frac{1}{1-cos ^{4} x} d x=-frac{1}{2 tan x}+ ) ( frac{k}{sqrt{2}} tan ^{-1}left(frac{tan x}{sqrt{2}}right)+C, ) where ( k= ) A ( frac{1}{2} ) B. ( -frac{1}{2} ) ( c cdot-1 ) D. | 12 |
| 855 | ( int sec x cdot log (sec x+tan x) d x= ) ( A cdot[log (sec x+tan x)]^{2}+c ) B. ( frac{[log (sec x+tan x)]^{2}}{2}+c ) c. ( -log (sec x+tan x)+c ) D. ( log (sec x+tan x)+c ) | 12 |
| 856 | x²+1 22. If f(x) = | e-dt, then f (x) increases in (a) (-2,2) (c) (0, 0) (b) no value of x (d) (-0,0) | 12 |
| 857 | ( int frac{cos x-sin x}{sqrt{8-sin 2 x}} d x ) | 12 |
| 858 | Evaluate: ( int_{3}^{9} frac{sqrt[3]{12-x}}{sqrt[3]{x}+sqrt[3]{12-x}} d x ) | 12 |
| 859 | Evaluate: ( int cot x log sin x d x ) A ( cdot(log sin x)^{2} ) B ( cdot frac{1}{2}(log sin x) ) c. ( frac{1}{2}(log operatorname{cosec} x)^{2} ) D. ( frac{1}{2}(log sin x)^{2} ) | 12 |
| 860 | Statement-1: The value of the integral T/3 dx tan r is equal to rt/6 [JEE M 2013] tolt ✓ Statement- (a +b – x)dx. (a) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1. Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (C) Statement-1 is true; Statement-2 is false. (d) Statement-1 is false; Statement-2 is true. | 12 |
| 861 | Integrate the function ( frac{e^{5 log x}-e^{4 log x}}{e^{3 log x}-e^{2 log x}} ) | 12 |
| 862 | Solve ( : int_{0}^{2} x sqrt{x+2} d x ) | 12 |
| 863 | Evaluate ( int frac{cos x-sin x}{cos x+sin x} cdot(2+ ) ( 2 sin 2 x) d x ) | 12 |
| 864 | Evaluate ( int_{0}^{2} frac{x}{3} d x ) | 12 |
| 865 | If ( boldsymbol{I}_{n}=int_{boldsymbol{pi} / 4}^{pi / 2}(boldsymbol{T} boldsymbol{a} boldsymbol{n} boldsymbol{theta})^{-boldsymbol{n}} cdot boldsymbol{d} boldsymbol{theta} ) for ( (boldsymbol{n}>1) ) then ( boldsymbol{I}_{boldsymbol{n}}+boldsymbol{I}_{boldsymbol{n}+mathbf{2}}=? ) A. ( frac{1}{mathrm{n}+1} ) B. ( frac{-1}{mathrm{n}+1} ) c. ( frac{1}{mathrm{n}-1} ) D. ( frac{-1}{mathrm{n}-1} ) | 12 |
| 866 | Using (i) or ( (i i) ) above the best upper bound of ( int_{0}^{1} sqrt{1+x^{4}} d x ) is A. ( 1+sqrt{2} ) B. ( frac{1+sqrt{2}}{2} ) c. ( frac{sqrt{2}-1}{2} ) D. ( 2(sqrt{2}-1) ) | 12 |
| 867 | ( int_{-1}^{1} e^{x} d x= ) | 12 |
| 868 | et ( frac{boldsymbol{d} boldsymbol{f}(boldsymbol{x})}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{e}^{sin boldsymbol{x}}}{boldsymbol{x}}, boldsymbol{x}>mathbf{0} ) ( int_{1}^{4} frac{3 e^{sin x^{3}}}{x} d x=f(k)-f(1) ) then one of the possible values of ( k ) is A . 16 B. 63 c. 64 D. 15 | 12 |
| 869 | Find ‘c’, so that ( boldsymbol{f}^{prime}(boldsymbol{c})=frac{boldsymbol{f}(boldsymbol{b})-boldsymbol{f}(boldsymbol{a})}{boldsymbol{b}-boldsymbol{a}} ) ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} ) at ( boldsymbol{a}=mathbf{0}, boldsymbol{b}=mathbf{1} ) | 12 |
| 870 | ( boldsymbol{n} stackrel{boldsymbol{L} t}{rightarrow} infty sum_{r=0}^{boldsymbol{n}-1} frac{1}{sqrt{boldsymbol{n}^{2}-boldsymbol{r}^{2}}} ) ( A ) B . ( pi / 2 ) c. ( pi / 3 ) D . ( pi / 6 ) | 12 |
| 871 | Integrate ( intleft(sin ^{-1} xright)^{2} d x ) | 12 |
| 872 | ( int_{0}^{pi / 4} tan ^{2} x d x= ) A ( cdot 1-frac{pi}{4} ) B ( cdot 1+frac{pi}{4} ) c. ( frac{-pi}{4}-1 ) D. ( frac{pi}{4}-1 ) | 12 |
| 873 | Evaluate the given integral: ( int_{0}^{5} x^{4} d x ) | 12 |
| 874 | Find ( int sqrt{1+cos 2 x} d x ) | 12 |
| 875 | Solve: ( int frac{cos x}{1+cos x} ) | 12 |
| 876 | ( int frac{boldsymbol{a}}{boldsymbol{b}+boldsymbol{c} boldsymbol{e}^{boldsymbol{x}}} boldsymbol{d} boldsymbol{x} ) A ( cdot frac{a}{b}left[x-log left(b+c e^{x}right)right] ) B ( cdot frac{a}{b}left[x+log left(b+c e^{x}right)right] ) c. ( frac{a}{b}left[x-log left(c e^{x}right)right] ) D. ( frac{a}{c}left[x+log left(b+c e^{x}right)right] ) | 12 |
| 877 | ( int_{1}^{2} e^{x}left(frac{1}{x}-frac{1}{x^{2}}right) d x ) equals to A ( cdot eleft(frac{e}{2}-1right) ) B. 1 ( c cdot e(e-1) ) D. ( frac{e}{2} ) | 12 |
| 878 | The value of ( int_{-pi / 2}^{pi / 2} log left(frac{2-sin theta}{2+sin theta}right) d theta ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. None of these | 12 |
| 879 | ( int frac{x+sin x}{1+cos x} d x= ) A ( cdot x tan frac{x}{2}+c ) B. ( x cot frac{x}{2}+c ) c. ( x sin frac{x}{2}+c ) D. ( x cos frac{x}{2}+c ) | 12 |
| 880 | ( int_{0}^{pi / 2} sin ^{5} x cos ^{6} x d x= ) A ( cdot frac{8}{693} ) в. ( frac{32}{693} ) c. ( frac{8}{99} ) D. ( frac{16}{63} ) | 12 |
| 881 | Evaluate the following definite integral: ( int_{1}^{2} frac{1}{x} d x ) | 12 |
| 882 | ( lim _{n rightarrow infty}left[frac{1}{3 n+1}+frac{1}{3 n+2}+dots+right. ) ( left.frac{1}{3 n+n}right] ) A ( cdot log (2 / 3) ) and 5 B. ( log (3 / 2) ) ( c cdot log (4 / 3) ) ( mathbf{D} cdot log (3 / 4) ) | 12 |
| 883 | 12. The value of 12004 | 11-r2 dx is 14 | 12 |
| 884 | The value of ( int e^{ln sqrt{x}} d x ) is | 12 |
| 885 | ( int x^{3}(log x)^{2} d x=frac{x^{4}}{4}(log x)^{2} ) ( frac{1}{8} x^{4} log x+frac{1}{4 k} x^{4} . ) Find the value of ( k ) | 12 |
| 886 | Evaluate: ( int_{0}^{pi / 4} sec ^{7} theta sin ^{3} theta d theta= ) A ( cdot frac{1}{12} ) в. ( frac{3}{12} ) c. ( frac{5}{12} ) D. ( frac{7}{12} ) | 12 |
| 887 | Solve : ( int_{-2}^{2}|2 x+3| d x ) | 12 |
| 888 | Evaluate the following integral: ( int_{-1}^{1}|2 x+1| d x ) | 12 |
| 889 | Evaluate ( int_{0}^{2} e^{x} d x ) as a limit of sum. | 12 |
| 890 | ( f int frac{cos x-sin x}{sqrt{8-sin 2 x}} d x= ) ( sin ^{-1}left(frac{boldsymbol{A}}{mathbf{4} mathbf{2 6}}(sin boldsymbol{x}+cos boldsymbol{x})right)+boldsymbol{C} ) then A is equal to | 12 |
| 891 | ( int(log x)^{2} d x ) | 12 |
| 892 | Evaluate ( int_{1}^{2} frac{2}{x} d x ) | 12 |
| 893 | ( int frac{3 x^{2}}{sqrt{left(9-16 x^{6}right)}} d x ) A. ( frac{1}{4} sin ^{-1} frac{4}{3} x^{3} ) B ( cdot frac{1}{2} sin ^{-1} frac{4}{3} x^{3} ) c. ( frac{1}{4} sin ^{-1} frac{2}{3} x^{3} ) D. ( frac{1}{4} sin ^{-1} frac{4}{3} x^{6 .} ) | 12 |
| 894 | ( intleft(frac{x^{6}-1}{x^{2}+1}right) d x ) | 12 |
| 895 | Evaluate the integral ( int_{0}^{pi} frac{d x}{a+b cos x} ) where ( a>b ) A ( cdot pi sqrt{a^{2}-b^{2}} ) an ( pi sqrt{a^{2}-b^{a^{2}}} ) в. ( pi a b ) c. ( frac{pi}{sqrt{a^{2}+b^{2}}} ) D. ( frac{pi}{sqrt{a^{2}-b^{2}}} ) | 12 |
| 896 | eosx 37. Integrate o e osx + e-cos x ax. | 12 |
| 897 | Integrate the following function with respect to ( x ) ( frac{sec ^{3} x}{csc x} ) | 12 |
| 898 | If ( boldsymbol{M}=int_{0}^{pi / 2} frac{cos boldsymbol{x}}{boldsymbol{x}+mathbf{2}} boldsymbol{d} boldsymbol{x}, boldsymbol{N}= ) ( int_{0}^{pi / 4} frac{sin x cos x}{(x+1)^{2}} d x quad, ) then the value of ( M-N ) is ( ? ) ( A ) в. c. ( frac{2}{pi-4} ) D. ( frac{2}{pi+4} ) | 12 |
| 899 | ( int frac{d x}{sqrt{(x-a)(b-x)}} ) equals A ( cdot sin ^{-1} sqrt{left(frac{x+a}{b-a}right)}+c ) B. ( cos ^{-1} sqrt{left(frac{x+a}{b-a}right)}+c ) ( ^{mathrm{c}} 2 sin ^{-1} sqrt{left(frac{x+a}{b-a}right)}+c ) D. None of these | 12 |
| 900 | Evaluate ( : int_{-2}^{2}|2 x+3| d x ) | 12 |
| 901 | Evaluate ( int frac{d x}{sqrt{x}(1+sqrt{x})} ) A ( cdot log (1+x)+c ) B. ( 2 log (x)+c ) c. ( 6 log (1+sqrt{x^{2}})+c ) D. ( 2 log (1+sqrt{x})+c ) | 12 |
| 902 | The value of ( int frac{log x}{(x+1)^{2}} d x ) is A ( cdot frac{-log x}{x+1}+log x-log (x+1)+C ) B. ( frac{log x}{x+1}+log x-log (x+1)+C ) c. ( frac{log x}{x+1}-log x-log (x+1)+C ) D. ( frac{-log x}{x+1}-log x-log (x+1)+C ) | 12 |
| 903 | ( frac{boldsymbol{x}+mathbf{2}}{boldsymbol{x}^{boldsymbol{3}}-boldsymbol{x}}= ) A. ( frac{1}{2(x+1)}+frac{3}{2(x-1)}-frac{2}{x} ) B. ( frac{1}{2(x+1)}-frac{3}{2(x-1)}-frac{2}{x} ) c. ( frac{1}{2(x+1)}-frac{3}{2(x-1)}+frac{2}{x} ) D. ( frac{1}{2(x+1)}+frac{3}{2(x-1)}+frac{2}{x} ) | 12 |
| 904 | If ( int frac{1}{(x+2)left(x^{2}+1right)} d x= ) ( a log left|1+x^{2}right|+b tan ^{-1} x+ ) ( frac{1}{5} log |x+2|+C ) A. ( a=-frac{1}{10} b=-frac{2}{5} ) B. ( a=frac{1}{10} b=-frac{2}{5} ) c. ( a=-frac{1}{10} b=frac{2}{5} ) D. ( a=frac{1}{10} b=frac{2}{5} ) | 12 |
| 905 | ( intleft(frac{x^{6}-1}{x^{2}+1}right) d x ) | 12 |
| 906 | Evaluate the following definite integral: ( int_{0}^{pi / 2} cos ^{2} x d x ) | 12 |
| 907 | ( int frac{1}{1+x^{3}} d x= ) A ( cdot frac{1}{3} log |x+1|-frac{1}{6} log left|x^{2}-x+1right|+ ) ( quad frac{1}{sqrt{3}} tan ^{-1}left(frac{2 x-1}{sqrt{3}}right)+c ) B ( cdot frac{1}{3} log |x+1|+frac{1}{6} log left|x^{2}-x+1right|+ ) ( quad frac{1}{sqrt{3}} tan ^{-1}left(frac{2 x-1}{sqrt{3}}right)+c ) c. ( frac{1}{3} log |x+1|-frac{1}{6} log left|x^{2}-x+1right|- ) ( quad frac{1}{sqrt{3}} tan ^{-1}left(frac{2 x-1}{sqrt{3}}right)+c ) D. ( -frac{1}{3} log |x+1|+frac{1}{6} log left|x^{2}-x+1right|+ ) ( quad frac{1}{sqrt{3}} tan ^{-1}left(frac{2 x-1}{sqrt{3}}right)+c ) | 12 |
| 908 | 14. Tete Let f:R (0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)? (JEE Adv. 2017) (a) x9-f(x) (b) x-12 *f(t)cost dt @ ef -S* f(t)sintdt (d) f(x) + Są f(t) sint dt | 12 |
| 909 | Solve ( : int_{0}^{1} cot ^{-1}left(1+x+x^{2}right) d x ) | 12 |
| 910 | ( int e^{a x} cdot sin (b x+c) d x ) | 12 |
| 911 | ITICS 14. If f() = {** sin x, for lets 2. then Š S(@dx = otherwise, -2 (a) I (b) i (c) 2 (2000) (2) 3 | 12 |
| 912 | Evaluate: ( int_{-1}^{1} x e^{x^{2}} d x ) | 12 |
| 913 | Find : ( int frac{2}{(1-x)left(1+x^{2}right)} d x ) | 12 |
| 914 | Integrate ( int frac{d x}{x^{4}+1} ) | 12 |
| 915 | Evaluate: ( int frac{boldsymbol{d t}}{left(1-t^{2}right)left(1-2 t^{2}right)} ) | 12 |
| 916 | Solve ( int frac{1}{sin x cos ^{3} x} d x ) | 12 |
| 917 | ( intleft(sec ^{2} x+csc ^{2} xright) d x ) | 12 |
| 918 | ( int_{0}^{pi / 2} frac{sin x-cos x}{1+sin x cos x} d x ) | 12 |
| 919 | ( int frac{cos x-sin x}{7-9 sin 2 x} d x ) A. ( frac{1}{24} ln frac{(4+3 sin x+3 cos x)}{(4-3 sin x-3 cos x)}+c ) B. ( frac{1}{24} ln frac{(4-3 cos x-3 sin x)}{(4-3 cos x+3 sin x)}+c ) c. ( frac{1}{12} ln frac{(4+3 sin x-3 cos x)}{(4-3 sin x+3 cos x)}+c ) D. ( frac{1}{12} ln frac{(4+3 cos x+3 sin x)}{(4-3 sin x-3 cos x)}+c ) | 12 |
| 920 | Solve ( int_{0}^{h} x(h-x) d x ) ( ^{A} cdot_{I}=frac{h^{3}}{3} ) в. ( _{I=} frac{h^{3}}{6} ) ( ^{mathrm{C}} cdot_{I}=-frac{h^{3}}{6} ) D. None of these | 12 |
| 921 | ( int_{0}^{pi} x f(sin x) d x ) is equal to ( mathbf{A} cdot pi int_{0}^{x} f(cos x) d x ) ( mathbf{B} cdot pi int_{0}^{x} f(sin x) d x ) ( ^{mathbf{C}} cdot frac{pi}{2} int_{0}^{x / 2} f(sin x) d x ) D ( cdot pi int_{0}^{pi / 2} f(cos x) d x ) | 12 |
| 922 | Integrate the following functions with espect to ( x: int frac{d x}{4 x+5} ) This question has multiple correct options A ( cdot frac{1}{4} ln (4 x+5)+c ) B. ( frac{1}{4} ln (4 x+5)-c ) ( ^{mathbf{c}} cdot frac{-1}{4} ln (4 x+5)-c ) D. ( 4 ln (4 x-5)-c ) | 12 |
| 923 | ( int_{pi / 6}^{pi / 3} frac{d x}{1+sqrt{tan x}} ) is equal to A ( cdot frac{pi}{12} ) в. ( frac{pi}{2} ) c. D. | 12 |
| 924 | ( x^{2} sqrt{1-x^{2}} d x= ) A ( cdot frac{1}{8} arcsin x-frac{1}{8} xleft(1-2 x^{2}right) sqrt{1-x^{2}}+C ) B ( cdot frac{1}{8} arcsin x+frac{1}{8} xleft(1-2 x^{2}right) sqrt{1+x^{2}}+C ) C ( cdot frac{1}{8} arcsin x+frac{1}{8} xleft(1-x^{2}right) sqrt{1-x^{2}}+C ) D. None of these | 12 |
| 925 | Evaluate the following definite integrals ( int_{0}^{1} frac{1}{1+x^{2}} d x ) | 12 |
| 926 | If ( int sin ^{-1} x cos ^{-1} x d x= ) ( boldsymbol{f}^{-1}(boldsymbol{x})left[frac{boldsymbol{pi}}{2} boldsymbol{x}-boldsymbol{x} boldsymbol{f}^{-1}(boldsymbol{x})-boldsymbol{2} sqrt{1-boldsymbol{x}^{2}}right] frac{boldsymbol{pi}}{boldsymbol{2}} ) ( 2 x+C, ) then A ( cdot f(x)=sin x ) B . ( f(x)=cos x ) c. ( f(x)=tan x ) D. None of these | 12 |
| 927 | Evaluate : ( int_{0}^{pi / 2} frac{sin x}{1+cos ^{2} x} d x ) | 12 |
| 928 | If ( I=int frac{1}{e^{x}} tan ^{-1}left(e^{x}right) d x, ) then I equals A ( cdot-e^{-x} tan ^{-1}left(e^{x}right)+log left(1+e^{2 x}right)+C ) B. ( x-e^{-x} tan ^{-1} e^{x}-frac{1}{2} log left(1+e^{x}right)+C ) c. ( x-e^{-x} tan ^{-1}left(e^{x}right)-frac{1}{2} log left(1+e^{2 x}right)+C ) D. none of these | 12 |
| 929 | Evaluate ( int frac{cos x}{(2+sin x)(3+4 sin x)} d x ) | 12 |
| 930 | Solve ( int frac{x^{3}+4 x^{2}-7 x+5}{x+2} d x ) | 12 |
| 931 | ( int_{pi / 4}^{3 pi / 4} frac{d x}{1+cos x} ) is equal to ( A cdot 2 ) B. -2 ( c cdot 1 / 2 ) D. ( -1 / 2 ) | 12 |
| 932 | ( mathrm{f} f_{0}^{pi / 3} frac{cos }{3+4 sin x} d x= ) ( K log frac{(3+2 sqrt{3})}{3} ) then ( K ) is A ( cdot frac{1}{2} ) B. ( c cdot frac{1}{4} ) D. | 12 |
| 933 | 25. Determi Determine a positive integer n < 5, such that e* (x – 1)" dx = 16-6e (1992 – 4 | 12 |
| 934 | ( int frac{d x}{left(1+x^{2}right)^{2}} ) | 12 |
| 935 | Evaluate ( int_{0}^{2}left(3 x^{2}-2right) d x ) | 12 |
| 936 | IF ( f(x)=x^{2} ) for ( 0 leq x leq 1, sqrt{x} ) for ( 1 leq ) ( x leq 2 ) then ( int_{0}^{2} f(x) d x= ) A ( cdot frac{4 sqrt{2}}{3} ) B. ( frac{4 sqrt{2}-1}{3} ) c. ( frac{sqrt{2}}{3} ) D. | 12 |
| 937 | What is ( int_{0}^{1} frac{tan ^{-1} x}{1+x^{2}} d x ) equal to ( ? ) A ( cdot frac{pi}{4} ) в. c. ( frac{pi^{2}}{8} ) D. ( frac{pi^{2}}{32} ) | 12 |
| 938 | Solve: ( int frac{d x}{left(2 x^{2}+3right)left(x^{2}-4right)} ) | 12 |
| 939 | Evaluate the integral ( int_{-3}^{3} log (sqrt{x^{2}+1}+x) d x= ) ( mathbf{A} cdot mathbf{0} ) B. ( log 2 ) c. ( -log 2 ) D. ( 2 log 2 ) | 12 |
| 940 | 2. If FO)=e”. 80) = x, y>0 and -y)g(y)dy, then [2003] (a) F(t) = te (b) F(t) =1-te’ (1+t) (©) F(t) = e’ -(1+t) (d) F(t) = te’. | 12 |
| 941 | Evaluate ( : quad I=int frac{x+9}{x^{2}+5} d x ) | 12 |
| 942 | Evaluate: ( int_{0}^{2}left(x^{2}+3right) d x ) as limit of sums | 12 |
| 943 | Trs 23. Evaluate ) *sin 2x sin coun) are o Evaluate – dx a 2 x – 1 | 12 |
| 944 | d t is (2010) 31. The value of lim li 1x³0x² 4+4 | 12 |
| 945 | Evaluate the following as the limit of sum : ( int_{0}^{2}(x+4) d x ) A .4 B. 6 c. 8 D. 10 | 12 |
| 946 | evaluate : [ boldsymbol{I}=int frac{2 x}{x^{2}-60 x+6} d x ] | 12 |
| 947 | Solve the differential equation: ( frac{d y}{d x}=frac{x^{2}-y^{2}}{2 x y} ) | 12 |
| 948 | ( int frac{d x}{9+16 sin ^{2} x} ) is equal to A ( cdot frac{1}{3} tan ^{-1}left(frac{3 tan x}{5}right)+c ) B ( cdot frac{1}{5} tan ^{-1}left(frac{tan x}{15}right)+c ) c. ( frac{1}{15} tan ^{-1}left(frac{tan x}{5}right)+c ) D. ( frac{1}{15} tan ^{-1}left(frac{5 tan x}{3}right)+c ) | 12 |
| 949 | 16. Evaluate ſ log[V1- x + V1+x]dx | 12 |
| 950 | Evaluate the following definite integral: ( int_{1}^{2} e^{2 x}left(frac{1}{x}-frac{1}{2 x^{2}}right) d x ) | 12 |
| 951 | The domain of ( sin (cos theta) ) A. ( z ) в. ( R ) ( c cdot Q ) D. ( N ) | 12 |
| 952 | Evaluate: ( intleft(frac{1}{(ln x)}-frac{1}{(ln x)^{2}}right) d x ) equals | 12 |
| 953 | Solve: ( lim _{n rightarrow infty}left{frac{1}{n+1}+frac{1}{n+2}+ldots+frac{1}{2 n}right}= ) ( A cdot log 2 ) B . ( log 3 ) ( c ) D. ( frac{pi}{2} ) | 12 |
| 954 | Let f(x) = x -[x], for every real number x, where x is + integral part of x. Then ‘ f(x) dx is (1998 – 2 Marks (a) 1 (6) 2 (c) o n (d) 1/2 | 12 |
| 955 | ( int frac{2 x-1}{2 x^{2}+2 x+1} d x= ) ( mathbf{A} cdot frac{1}{2} ln left|2 x^{2}+2 x+1right|+2 tan ^{-1}(2 x+1)+c ) B ( cdot-frac{1}{2} ln left|2 x^{2}+2 x+1right|-2 tan ^{-1}(2 x+1)+c ) C ( -frac{1}{2} ln left|2 x^{2}+2 x+1right|+2 tan ^{-1}(2 x+1)+c ) D ( cdot frac{1}{2} ln left|2 x^{2}+2 x+1right|-2 tan ^{-1}(2 x+1)+c ) | 12 |
| 956 | The value of ( int_{0}^{2 pi}|cos x-sin x| d x ) is equal to A ( cdot 2 sqrt{2} ) B. 2 ( c cdot 4 ) D. ( 4 sqrt{2} ) | 12 |
| 957 | Evaluate: ( int frac{1}{9 x^{2}+49} d x ) | 12 |
| 958 | Value of ( int frac{d x}{x^{2}left(x^{4}+1right)^{3 / 4}} ) is : A ( -left(1+frac{1}{x^{4}}right)^{frac{1}{4}}+c ) B. ( quadleft(1+frac{1}{x^{4}}right)^{frac{1}{4}}+c ) c. ( quad-left(1-frac{1}{x^{4}}right)^{frac{1}{4}}+c ) D. None of these | 12 |
| 959 | If ( boldsymbol{I}=int sqrt{frac{mathbf{5}-boldsymbol{x}}{mathbf{5}+boldsymbol{x}}} boldsymbol{d} boldsymbol{x}, ) then ( boldsymbol{I} ) equals A ( cdot 5 sin ^{-1}left(frac{x}{5}right)+sqrt{25-x^{2}}+C ) B. ( 10 sin ^{-1}left(frac{x}{5}right)+sqrt{25-x^{2}}+C ) c. ( 5 sin ^{-1}left(frac{x}{5}right)-sqrt{25-x^{2}}+C ) D. none of these | 12 |
| 960 | 8. If f(a+b – x) = f(x) then xf (x)dx is equal to [2003 @) at b j r(a + b + xwek (by a to provide (c) at bº f(x)dx (a) b-a; f(x)dx. а 2 2 a a no 1 | 12 |
| 961 | ( int frac{x^{2}}{left(x^{2}+2right)left(x^{2}+3right)} d x= ) ( mathbf{A} cdot-sqrt{2} tan ^{-1} x+sqrt{3} tan ^{1} x+c ) B ( cdot-sqrt{2} tan ^{-1}left(frac{x}{sqrt{2}}right)+sqrt{3} tan ^{-1}left(frac{x}{sqrt{3}}right)+c ) C ( cdot sqrt{2} tan ^{-1}left(frac{x}{sqrt{2}}right)+sqrt{3} tan ^{-1}left(frac{x}{sqrt{3}}right)+c ) D. None of these | 12 |
| 962 | Find ( int frac{d x}{xleft(x^{3}+1right)^{2}} ) | 12 |
| 963 | 6. 5 tan x If the dx = x +aln sin x – 2 cos x +k, then ais J tan x-2 equal to : [2012] (a) -1 (b) -2 (c) 1 (d) 2 | 12 |
| 964 | Integrate ( int frac{x}{x^{2}+x+1} d x ) | 12 |
| 965 | ( int frac{1}{(x+2)(x+3)} d x ) | 12 |
| 966 | Integrate the function ( frac{1}{x-x^{3}} ) | 12 |
| 967 | ( int xleft(fleft(x^{2}right) g^{prime prime}left(x^{2}right)-f^{prime prime}left(x^{2}right) gleft(x^{2}right)right) d x= ) A ( cdot fleft(x^{2}right) g^{prime}left(x^{2}right)-gleft(x^{2}right) f^{prime}left(x^{2}right)+c ) B ( cdot frac{1}{2}left(fleft(x^{2}right) gleft(x^{2}right) f^{prime}left(x^{2}right)right)+c ) c. ( frac{1}{2}left(fleft(x^{2}right) g^{prime}left(x^{2}right)-gleft(x^{2}right) f^{prime}left(x^{2}right)right)+c ) D. none of these | 12 |
| 968 | If ( boldsymbol{f}(boldsymbol{x})= ) ( mid begin{array}{ccc}sin x+sin 2 x+sin 3 x & sin 2 x & sin 3 x \ 3+4 sin x & 3 & 4 sin x \ 1+sin x & sin x & 1end{array} ) then the value of ( int_{0}^{frac{pi}{2}} f(x) d x, ) is A . 3 B. ( frac{2}{3} ) ( c cdot frac{1}{3} ) D. | 12 |
| 969 | ( int frac{x}{x^{4}+x^{2}+1} d x, ) Integration gives ( frac{1}{sqrt{(k)}} tan ^{-1} frac{2 x^{2}+1}{sqrt{(k)}} ) find ( k^{2} ) | 12 |
| 970 | Solve: ( int frac{d x}{2 x^{2}+x-1} ) | 12 |
| 971 | ( int frac{boldsymbol{x}+sqrt[3]{boldsymbol{x}^{2}}+sqrt[6]{boldsymbol{x}}}{boldsymbol{x}(1+sqrt[3]{boldsymbol{x}})} boldsymbol{d} boldsymbol{x} ) is equal to A ( cdot frac{3}{2} x^{2 / 3}+6 tan ^{-1} x^{1 / 6}+c ) B. ( frac{3}{2} x^{2 / 3}-6 tan ^{-1} x^{1 / 6}+c ) c. ( -frac{3}{2} x^{2 / 3}+6 tan ^{-1} x^{1 / 6}+c ) D. None of these | 12 |
| 972 | Solve : ( int frac{x^{2}}{(4+x)^{3 / 2}} d x ) | 12 |
| 973 | ( int frac{boldsymbol{x} boldsymbol{T} boldsymbol{a} boldsymbol{n}^{-1} boldsymbol{x}}{left(1+boldsymbol{x}^{2}right)^{3 / 2}} boldsymbol{d} boldsymbol{x}= ) A. ( frac{x+operatorname{Tan}^{-1} x}{left(1+x^{2}right)^{3} / 2}+c ) в. ( frac{x-operatorname{Tan}^{-1} x}{sqrt{left(1+x^{2}right)}}+c ) c. ( frac{x}{sqrt{1+x^{2}}}-operatorname{Tan}^{-1} x+c ) D. ( frac{x}{1+x^{2}}+operatorname{Tan}^{-1} x+c ) | 12 |
| 974 | ( int_{0}^{pi / 4} sec ^{7} theta sin ^{3} theta d theta ) A ( .1 / 12 ) в. ( 3 / 12 ) c. ( 5 / 12 ) D. none of these | 12 |
| 975 | Evaluate ( int_{0}^{pi} frac{x}{a^{2} cos ^{2} x+b^{2} sin ^{2} x} d x ) | 12 |
| 976 | If ( int frac{boldsymbol{f}(boldsymbol{x})}{log (sin boldsymbol{x})} boldsymbol{d} boldsymbol{x}=log [log sin boldsymbol{x}]+boldsymbol{c} ) ( operatorname{then} f(x)=dots ) A . ( cot x ) B. ( tan x ) ( c . sec x ) D. ( operatorname{cosec} x ) | 12 |
| 977 | ( int 5^{5^{5^{x}}} cdot 5^{5^{x}} cdot 5^{x} d x ) is equal to | 12 |
| 978 | TU/sin x + cos x dx 6. Evaluate : J 9+16 sin 2x | 12 |
| 979 | Evaluate the following integrals: ( int_{0}^{pi} x d x ) | 12 |
| 980 | Solve ( int frac{1-sqrt{x}}{1+sqrt{x}} d x ) A ( cdot 3 sqrt{x}+frac{x}{2}-3 log (1+sqrt{x})+c ) B ( 3 sqrt{x}+3 log (1+sqrt{x})-frac{1}{2} x+c ) c. ( 3 sqrt{x}-frac{1}{2} x-3 log (1+sqrt{x})++c ) D. ( 4 sqrt{x}-x-4 log (1+sqrt{x})+c ) | 12 |
| 981 | Resolve ( frac{2 x^{2}-11 x+5}{(x-3)left(x^{2}+2 x+5right)} ) into partial fractions. A ( frac{1}{2(x-3)}-frac{(5 x-5)}{2left(x^{2}+2 x+5right)} ) B. ( frac{1}{2(x-3)}+frac{(5 x-5)}{2left(x^{2}+2 x+5right)} ) C ( frac{1}{(x-3)}+frac{(5 x-5)}{left(x^{2}+2 x+5right)} ) D ( frac{1}{2(x+3)}+frac{(5 x-5)}{2left(x^{2}+2 x+5right)} ) | 12 |
| 982 | If ( frac{2 x+A}{(x-3)(x+2)}=frac{9}{5(x-3)}+ ) ( frac{B}{(x+2)}, ) then This question has multiple correct options ( mathbf{A} cdot A=3 ) B. ( B=5 ) c. ( _{A}=frac{1}{3} ) D. ( B=frac{1}{5} ) | 12 |
| 983 | Find: ( int frac{4}{(x-2)left(x^{2}+4right)} d x ) | 12 |
| 984 | Show that: ( int_{-a}^{a} f(x) d x=2 int_{0}^{a} f(x) d x, ) if ( f(x) ) is an even function. ( boldsymbol{I}=mathbf{0}, ) if ( boldsymbol{f}(boldsymbol{x}) ) is an odd function. | 12 |
| 985 | X 6. s(2 sin x + 4) dr is equal to dx is equal to (a) -2 cos x + log x+c (b) 2 cos x + log x + c (e) -2 sin x-*+c (d) -2 cos x +*+c | 12 |
| 986 | Solve: ( int_{0}^{1} frac{1}{2 x^{2}+x+1} d x ) | 12 |
| 987 | 22. If*f is a continous function with | f(t)dt = 0 as x1 then show that every line y=mx (0,/2) of X (x,0) ron BTO -√2) intersects the curve y2 + f(t)dt = 2! (1991 – 4 Marks) | 12 |
| 988 | The value of ( frac{(sqrt{mathbf{2}}+mathbf{1}) mathbf{1 9 8}}{boldsymbol{pi}} int_{boldsymbol{pi} / 4}^{boldsymbol{3} boldsymbol{pi} / boldsymbol{4}} frac{boldsymbol{phi}}{mathbf{1}+sin phi} boldsymbol{d} boldsymbol{phi} ) is | 12 |
| 989 | 5. S V1+ sin x dx = (a) }( sin + cos}+c (C) 2/1+sin x + c (d) -2/1-sin x + c | 12 |
| 990 | ( f(x-1)(x+2)(x-3)=A+ ) ( frac{B}{(2 x-1)}+frac{C}{(x+2)}+frac{D}{(x-3)} ) then ( mathbf{A}= ) A ( cdot frac{1}{2} ) B. ( frac{-1}{50} ) ( c cdot frac{-8}{25} ) D. ( frac{27}{25} ) | 12 |
| 991 | ( int e^{x} sqrt{1+e^{x}} d x= ) A ( cdotleft(1+e^{x}right)^{frac{3}{2}}+c ) B ( cdot frac{2}{3}left(1-e^{x}right)^{3 / 2}+c ) ( mathbf{c} cdotleft(1-e^{x}right)^{3 / 2}+c ) D. ( frac{2}{3}left(1+e^{x}right)^{3 / 2}+c ) | 12 |
| 992 | 21. If y = 3×2 + 2x + 4, then ſy dx will be… | 12 |
| 993 | 29. Let I = | sin* dx and J = 1 Cos* dx. Then which one of 0 the following is true? 2 (a) 1>and) >2 (2) I2 (b) I< and Iand J <2 | 12 |
| 994 | ( f(x)=left{begin{array}{cc}e^{cos x} cdot sin x & text { for }|x| leq 2 \ 2 & text { otherwise }end{array}right. ) then ( int_{-2}^{3} f(x) d x ) is equal to A . B. ( c cdot 2 ) ( D ) | 12 |
| 995 | The question is refer to image. 8 | 12 |
| 996 | The integral ( int_{pi / 12}^{pi / 4} frac{8 cos 2 x}{(tan x+cot x)^{3}} d x ) equals: A ( cdot frac{15}{128} ) в. ( frac{15}{64} ) c. ( frac{13}{32} ) D. ( frac{13}{256} ) | 12 |
| 997 | If ( int frac{2^{x}}{sqrt{1-4^{x}}} d x=K sin ^{-1}left(2^{x}right)+C ) then ( K ) is equal to A ( . e n 2 ) в. ( frac{1}{2} ell n 2 ) ( c cdot frac{1}{2} ) D. ( frac{1}{ell n^{2}} ) | 12 |
| 998 | ( int sin ^{3} x cos ^{2} x d x ) is equal to | 12 |
| 999 | 5. Evaluate the following | 204+1)3/4 | 12 |
| 1000 | ( int_{0}^{1} frac{e^{x}}{1+e^{2 x}} d x ) | 12 |
| 1001 | The value of the integral ( int_{frac{pi}{6}}^{frac{pi}{2}}left(frac{1+sin 2 x+cos 2 x}{sin x+cos x}right) d x ) is equal to A . 16 B. 8 ( c cdot 4 ) ( D ) | 12 |
| 1002 | ( int frac{1}{x+x log x} d x ) | 12 |
| 1003 | If ( frac{2 x^{2}+3 x+4}{(x-1)left(x^{2}+2right)}=frac{A}{x-1}+frac{B x+C}{x^{2}+2} ) Then the value of ( B ) is equal to A . 3 B. – ( c cdot-2 ) D. | 12 |
| 1004 | Evaluate ( int_{0}^{2}left(x^{2}-xright) d x ) | 12 |
| 1005 | Ven3 xsin x2 33. The value of vino sin x + sin(ln6 – x2) I dx is (2011) a broma con ben con contest NIw (0) Come | 12 |
| 1006 | Let ( a, b, c ) be such that [ frac{1}{(1-x)(1-2 x)(1-3 x)}=frac{a}{1-x}+ ] ( frac{b}{1-2 x}+frac{c}{1-3 x} ) then ( frac{a}{1}+frac{b}{3}+frac{c}{5}= ) A ( cdot frac{1}{15} ) B. ( frac{1}{6} ) ( c cdot frac{1}{5} ) D. 1 ( overline{3} ) | 12 |
| 1007 | Integrate: ( int frac{1}{1+tan x} d x ) | 12 |
| 1008 | Evaluate the following integral by expressing them as a limit of a sum. ( int_{1}^{2}(3 x-2) d x ) A ( cdot frac{1}{2} ) B. ( frac{3}{2} ) ( c cdot frac{5}{2} ) D. ( frac{7}{2} ) | 12 |
| 1009 | Evaluate the following integral ( int frac{1}{sqrt{a^{2}+b^{2} x^{2}}} d x ) | 12 |
| 1010 | ( boldsymbol{I}_{mathrm{n}}=int_{1}^{mathrm{e}}(log mathrm{x})^{mathrm{n}} mathrm{d} mathbf{x} ) and ( mathbf{I}_{mathrm{n}}=mathbf{A}+mathbf{B I}_{mathbf{n}-mathbf{1}} ) then ( mathbf{A}=dots dots dots dots . . quad B=dots . . . . . . . . . . . . . . ) ( mathbf{A} cdot e,-n ) B. ( 1 / e, n ) c. ( -e, n ) D. ( -e ) | 12 |
| 1011 | ( int e^{x / 2} sin left(frac{pi}{4}+frac{x}{2}right) d x= ) A ( cdot sqrt{2} e^{x / 2} sin frac{x}{2}+c ) B . ( sqrt{2} e^{x / 2} cos frac{x}{2}+c ) C ( cdot-sqrt{2} e^{x / 2} sin frac{x}{2}+c ) D. ( -sqrt{2} e^{x / 2} cos frac{x}{2}+c ) | 12 |
| 1012 | ( int frac{e^{x}}{sqrt{5-4 e^{x}+e^{2 x}}} d x ) ( mathbf{A} cdot cos ^{-1}left(frac{e^{x}+2}{3}right)+c ) ( mathbf{B} cdot cos ^{-1}left(frac{e^{x}-3}{2}right)+c ) ( mathbf{C} cdot sin ^{-1}left(frac{e^{x}+2}{3}right)+c ) ( mathbf{D} cdot sin ^{-1}left(frac{e^{x}-3}{2}right)+c ) | 12 |
| 1013 | Evaluate the following integral: ( int frac{left(e^{sin ^{-1} x}right)^{2}}{sqrt{1-x^{2}}} d x ) | 12 |
| 1014 | Evaluate the integral ( int_{0}^{pi} x sin ^{5} x cos ^{6} x d x=? ) A ( cdot frac{5 pi}{16} ) в. ( frac{35 pi}{128} ) c. ( frac{5 pi}{8} ) D. ( frac{8 pi}{693} ) | 12 |
| 1015 | Evaluate the following integral: ( int_{0}^{2}(3 x+2) d x ) | 12 |
| 1016 | The value of ( sum_{r=1}^{n} int_{0}^{1} f(r-1+x) d x ) is equal to (if function has period 1 ) A ( cdot int_{0}^{1} f(x) d x ) B. ( _{n int_{0}}^{1} f(x) d x ) c. ( (n-1) int_{0}^{1} f(x) d x ) D. ( int_{0}^{n} f(x) d x ) | 12 |
| 1017 | Integrate the rational function ( frac{3 x-1}{(x-1)(x-2)(x-3)} ) | 12 |
| 1018 | If ( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{left(boldsymbol{x}^{2}+boldsymbol{a}^{2}right)left(boldsymbol{x}^{2}+boldsymbol{b}^{2}right)left(boldsymbol{x}^{2}+boldsymbol{c}^{2}right)} ) then ( I ) equals A ( cdot frac{1}{b c} tan ^{-1}(a)+frac{1}{c a} tan ^{-1}(b)+frac{1}{c b} tan ^{-1}(c)+k ) B. ( frac{1}{b^{2}-c^{2}} tan ^{-1}(a)+frac{1}{c^{2}-a^{2}} tan ^{-1}(b)+ ) ( frac{1}{a^{2}-b^{2}} tan ^{-1}(c)+k ) ( frac{tan ^{-1} a+tan ^{-1} b+tan ^{-1} c}{a^{2}+b^{2}+c^{2}}+k ) D. none of these | 12 |
| 1019 | ( I=int_{0}^{pi / 2} frac{x sin x cos x}{cos ^{4} x+sin ^{4} x} d x ) ( therefore boldsymbol{I}=boldsymbol{pi}^{2} / boldsymbol{k} ) what is k? | 12 |
| 1020 | State whether the given statement is True or False ( int_{0}^{2} e^{x} d x ) can be represented as ( 2 lim _{n rightarrow infty} frac{1}{n}left[e^{0}+e^{frac{2}{n}}+e^{frac{4}{n}}+ldots ldots+e^{frac{2(n-1)}{n}}right] ) A. True B. False | 12 |
| 1021 | ( int frac{x}{x^{4}-1} d x ) | 12 |
| 1022 | 43. If y6 = I can see yo tothen find, x= 43. cos x cos Jo If y(x) = -do, then find 215 ² 116 at | 12 |
| 1023 | Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+boldsymbol{phi}(boldsymbol{x}) ) where ( phi(boldsymbol{x}) ) is an even function then find the value of ( int_{-1}^{1} x f(x) d x ) | 12 |
| 1024 | ( int frac{sin 2 x}{sin ^{2} x+2 cos ^{2} x} d x= ) A ( cdot log left(1+cos ^{2} xright)+C ) В ( cdot log left(1+tan ^{2} xright)+C ) ( mathbf{c} cdot-log left(1+sin ^{2} xright)+C ) ( mathbf{D} cdot-log left(1+cos ^{2} xright)+C ) | 12 |
| 1025 | ( int_{0}^{1} tanh x d x= ) ( A cdot log (e+1 / e) ) B. ( log (e-1 / e) ) ( mathbf{c} cdot log (mathrm{e} / 2+1 / 2 mathrm{e}) ) D. ( log left(frac{e}{2}-frac{1}{e}right) ) | 12 |
| 1026 | sin x 1. Evaluate dx sin x – COS X | 12 |
| 1027 | Evaluate: ( int frac{2 x}{(x+5)^{2}} d x ) | 12 |
| 1028 | Find ( int frac{boldsymbol{d} boldsymbol{x}}{sqrt{mathbf{2} boldsymbol{x}-boldsymbol{x}^{2}}} ) | 12 |
| 1029 | ( int_{1}^{e} log x d x=_{-} ) A ( cdot e+1 ) в. ( e-1 ) ( mathbf{c} cdot e+2 ) ( D ) | 12 |
| 1030 | ( frac{boldsymbol{x}^{boldsymbol{4}}-mathbf{5} boldsymbol{x}^{boldsymbol{2}}+mathbf{1}}{left(boldsymbol{x}^{mathbf{2}}+mathbf{1}right)^{mathbf{3}}}= ) ( mathbf{A} cdot frac{1}{left(mathbf{x}^{2}+1right)}-frac{1}{left(mathbf{x}^{2}+1right)^{2}}+frac{7}{left(mathbf{x}^{2}+1right)^{3}} ) ( mathbf{B} cdot frac{1}{left(mathbf{x}^{2}+1right)}-frac{7}{left(mathbf{x}^{2}+1right)^{2}}+frac{7}{left(mathbf{x}^{2}+1right)^{3}} ) ( mathbf{C} cdot frac{7}{left(mathbf{x}^{2}+1right)}-frac{7}{left(mathbf{x}^{2}+1right)^{2}}+frac{1}{left(mathbf{x}^{2}+1right)^{3}} ) ( frac{7}{left(mathbf{x}^{2}+1right)}-frac{1}{left(mathbf{x}^{2}+1right)^{2}}-frac{1}{left(mathbf{x}^{2}+1right)^{3}} ) | 12 |
| 1031 | ( int frac{3 sin x+2 cos x}{sin x+cos x} d x ) | 12 |
| 1032 | Evaluate : ( int frac{x+1}{x^{2}+3 x+12} d x ) | 12 |
| 1033 | ( int frac{1}{sqrt{e^{5 x}, sqrt[4]{left(e^{2 x}+e^{-2 x}right)^{3}}}} d x ) ( mathbf{A} cdot-t^{1 / 4}, ) where ( 1+e^{-4 x}=t ) B. ( t^{1 / 4}, ) where ( 1+e^{-4 x}=t ) ( mathbf{c} cdot-t^{3 / 4}, ) where ( 1+e^{-4 x}=t ) ( mathbf{D} cdot-t^{1 / 2}, ) where ( 1+e^{-4 x}=t ) | 12 |
| 1034 | The value of ( int_{0}^{sqrt{2}}left[x^{2}right] d x, ) where [.] is the greatest integer function, is A ( .2-sqrt{2} ) B. ( 2+sqrt{2} ) c. ( sqrt{2}-1 ) D. ( sqrt{2}-2 ) | 12 |
| 1035 | ( lim _{boldsymbol{n} rightarrow infty} frac{mathbf{1}^{boldsymbol{p}}+boldsymbol{2}^{boldsymbol{p}}+ldots+boldsymbol{n}^{boldsymbol{p}}}{boldsymbol{n}^{boldsymbol{p}+1}} ) is A. ( frac{1}{p+1} ) B. ( frac{1}{1-p} ) c. ( frac{1}{p}-frac{1}{p-1} ) D. ( frac{1}{p+2} ) | 12 |
| 1036 | ( int frac{x^{2}+x-6}{(x-2)(x-1)} d x= ) | 12 |
| 1037 | 2. Integrate the following: S(2t – 4)4 dt | 12 |
| 1038 | Find the value of ( int sec ^{2} x tan ^{3} x d x ) | 12 |
| 1039 | Evaluate: ( int sec x tan x d x ) | 12 |
| 1040 | Prove that ( int_{0}^{a} f(x) d x=int_{0}^{a} f(a-x) d x ) and hence evaluate ( int_{0}^{a} frac{sqrt{x}}{sqrt{x}+sqrt{a-x}} d x ) | 12 |
| 1041 | 2. If FO)=e”. 80) = x, y>0 and -y)g(y)dy, then [2003] (a) F(t) = te (b) F(t) =1-te’ (1+t) (©) F(t) = e’ -(1+t) (d) F(t) = te’. | 12 |
| 1042 | ( int frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x} cos boldsymbol{theta}+1} cdot ) A. ( frac{1}{cos theta} tan ^{-1} frac{x-cos theta}{sin theta} ) s. ( frac{1}{sin theta} tan ^{-1} frac{1-cos theta}{sin theta} ) D. ( frac{1}{sin theta} tan ^{-1} frac{x-cos theta}{sin theta} ) ( frac{1}{sin theta} tan ^{-1} frac{x+cos theta}{sin theta} ) | 12 |
| 1043 | 32 25. The value of the integralſ, – X – dx 1 + x is (2004 (a) +1 (b) -1 0 -1 (d) 1 | 12 |
| 1044 | ( int sqrt{frac{x}{x-1}} d x, x in(0, pi / 2) ) equals A ( cdot sqrt{x(x-1)}+log (sqrt{x}+sqrt{x-1})+c ) B . ( sqrt{x(x-1)}-log (sqrt{x}+sqrt{x-1})+c ) c. ( sqrt{x(x-1)}+log (sqrt{x}-sqrt{x-1})+c ) D. ( sqrt{x(x+1)}-log (sqrt{x}-sqrt{x+1})+c ) | 12 |
| 1045 | Find the general solution of ( frac{d y}{d x}=frac{2 y}{x} ) A ( cdot y=e^{5 log x+c} ) B . ( y=e^{3 log x+c} ) C ( cdot y=e^{2 log x+c} ) D. None of these | 12 |
| 1046 | ( int frac{left(e^{2 x}-1right)}{e^{2 x}+1} d x ) ( mathbf{A} cdot log left(e^{x}+e^{-x}right)+C ) B ( cdot log left(e^{x}-e^{-x}right)+C ) ( mathbf{c} cdot log left(e^{2 x}+e^{-2 x}right)+C ) D. ( log left(e^{2 x}-e^{-2 x}right)+C ) | 12 |
| 1047 | 1. The value of the definite integral ‘) dx is (a) (c) – 1 1+e-1 (b) 2 (1981 – 2 Marks) (d) none of these | 12 |
| 1048 | Evaluate ( int_{0}^{2}|1-x| d x ) | 12 |
| 1049 | Evaluate the integral ( int_{pi / 4}^{pi / 2} log (1+cot x) d x ) A ( cdot frac{pi}{4} log 2 ) в. ( frac{pi}{8} log 2 ) c. ( pi log 2 ) D. | 12 |
| 1050 | Evaluate the following integral: ( int frac{x}{sqrt{4-x^{4}}} d x ) | 12 |
| 1051 | ( int e^{tan ^{-1} x}left(1+x+x^{2}right) dleft(cot ^{-1} xright) ) is equal to A ( cdot-e^{tan ^{-1} x}+c ) B – ( e^{tan ^{-1} x}+c ) C. ( _{-x e^{tan ^{-1} x}+c} ) D. ( x e^{tan ^{-1} x}+c ) | 12 |
| 1052 | Solve: ( int x^{2 / 3}left(1+x^{5 / 3}right)^{frac{-13}{3}} d x ) ( mathbf{A} cdot-frac{9}{25left(1+x^{5 / 3}right)^{frac{10}{3}}}+C ) B. ( -frac{9}{15left(1+x^{5 / 3}right)^{frac{10}{3}}}+C ) c. ( -frac{9}{50left(1+x^{5 / 3}right)^{frac{10}{3}}}+C ) D. None of these | 12 |
| 1053 | ( int frac{x+1}{xleft(1+x e^{x}right)} d x=0 ) ( ^{mathbf{A}} cdot log left|frac{1+x e^{x}}{x e^{x}}right|+C ) ( ^{mathbf{B}} cdot log left|frac{x e^{x}}{1+x e^{x}}right|+C ) ( mathbf{c} cdot log left|x e^{x}left(1+x e^{x}right)right|+C ) ( mathbf{D} cdot log left(1+x e^{x}right)+C ) | 12 |
| 1054 | 42. Iffis an even function then prove that (2003 – 2 Mark Tt/2 I f(cos 2x) cos x dx = 12 f (sin 2x) cos x dx. T/4 | 12 |
| 1055 | 1/2 13. Value of cos 31 dt is ما را به زیبا | 12 |
| 1056 | If ( int frac{d x}{sqrt{x}(x+9)}=f(x)+ ) constant, then ( boldsymbol{f}(boldsymbol{x})= ) A ( cdot frac{2}{3} tan ^{-1} sqrt{x} ) B ( cdot frac{2}{3} tan ^{-1}left(frac{sqrt{x}}{3}right) ) C. ( tan ^{-1} sqrt{x} ) D. ( tan ^{-1}left(frac{sqrt{x}}{3}right) ) | 12 |
| 1057 | Solve ( int x sin ^{2} x d x ) A ( cdot frac{(x-1)}{2}left(x-frac{cos 2 x}{2}right)+C ) в. ( frac{(x-1)}{2}left(x-frac{sin 2 x}{2}right)+C ) c. ( frac{(x+1)}{2}left(x-frac{sin 2 x}{2}right)+C ) D. None of these | 12 |
| 1058 | ( int e^{x}left(frac{1+sin x}{1+cos x}right) d x ) is A ( cdot e^{x} tan left(frac{x}{2}right)+C ) B cdot ( tan left(frac{x}{2}right)+C ) ( mathbf{c} cdot e^{x}+C ) D. ( e^{x} sin x+C ) | 12 |
| 1059 | The value of ( int_{-pi / 2}^{pi / 2}left(operatorname{psin}^{3} x+q sin ^{4} x+r sin ^{5} xright) ) does not depend on ( A cdot p, q, r ) B. p, ronly c. ponly D. ( q ), ronly | 12 |
| 1060 | ( int frac{1}{x log x[log (log x)]} d x= ) ( mathbf{A} cdot log |log (log x)|+c ) ( mathbf{B} cdot log |log x|+c ) ( mathbf{c} cdot-log |log x|+c ) ( mathbf{D} cdot-log |log (log x)|+c ) | 12 |
| 1061 | Given Function ( f(x)== ) ( left{begin{array}{cc}x^{2}, & text { for } 0 leq x<1 \ sqrt{x}, & text { for } 1 leq x leq 2end{array}right} ) Evaluate ( int_{0}^{2} f(x) d x ) A ( cdot frac{1}{3}(4 sqrt{2}-1) ) B. ( frac{1}{3}(2 sqrt{2}-1) ) c. ( frac{2}{3}(4 sqrt{2}-1) ) D ( cdot frac{2}{3}(2 sqrt{2}-1) ) | 12 |
| 1062 | ( int_{0}^{pi / 2} f(sin 2 x) sin x d x= ) ( K int_{0}^{pi / 2} f(cos 2 x) cos x d x ) where ( k ) equals to A . 2 B. 4 ( c cdot sqrt{2} ) D. ( 2 sqrt{2} ) | 12 |
| 1063 | Evaluate ( int_{1}^{3}(3 x-2) d x ) | 12 |
| 1064 | Evaluate : ( int frac{1}{2 x^{2}+x+1} d x ) | 12 |
| 1065 | ( int frac{1}{left(1+x^{2}right) sqrt{left[p^{2}+q^{2}left(tan ^{-1} xright)^{2}right]}} d x ) A ( cdot frac{1}{q} log [t-sqrt{p^{2}+t^{2}}] ) where ( t=q tan ^{-1} x ) B. ( frac{1}{t} log [t+sqrt{p^{2}+t^{2}}] ) where ( t=q tan ^{-1} x ) c. ( frac{1}{q} log [t+sqrt{p^{2}+t^{2}}] ) where ( t=q tan ^{-1} x ) D. ( frac{1}{q} log [t+sqrt{p^{2}-t^{2}}] ) where ( t=q tan ^{-1} x ) | 12 |
| 1066 | ( I=int sec x tan x d x ) is equal to A. ( sec x+c ) B. ( cos x+c ) c. ( tan x+c ) D. None of these | 12 |
| 1067 | Let ( boldsymbol{f} ) be a function satisfying ( boldsymbol{f}^{prime prime}(boldsymbol{x})= ) ( x^{-3 / 2}, f^{prime}(4)=2 ) and ( f(0)=0 . ) Then ( f(784) ) is equal to | 12 |
| 1068 | If ( int x frac{ln (x+sqrt{1+x^{2}})}{sqrt{1+x^{2}}} d x= ) ( boldsymbol{a} sqrt{mathbf{1}+boldsymbol{x}^{2}} ln (boldsymbol{x}+sqrt{mathbf{1}+boldsymbol{x}^{2}})+boldsymbol{b} boldsymbol{x}+boldsymbol{c} ) then A ( . a=1, b=-1 ) В. ( a=1, b=1 ) c. ( a=-1, b=1 ) | 12 |
| 1069 | ( int_{0}^{2 pi}(sin x+|sin x|) d x ) is equal to A . B. 4 ( c cdot 8 ) D. | 12 |
| 1070 | ( int sqrt{e^{2 x}-1} d x ) is equal to | 12 |
| 1071 | If ( I_{n}=int_{0}^{pi / 4} tan ^{n} x d x ) then ( lim _{n rightarrow infty} nleft(I_{n}+right. ) ( left.boldsymbol{I}_{n-2}right)= ) ( mathbf{A} cdot mathbf{1} ) B. 1/2 ( c cdot infty ) D. | 12 |
| 1072 | Solve ( intleft(3 x^{2}-4right) x d x, x in R ) | 12 |
| 1073 | ( operatorname{Let} boldsymbol{I}=int_{boldsymbol{pi} / 4}^{pi / 3} frac{sin boldsymbol{x}}{boldsymbol{x}} boldsymbol{d} boldsymbol{x} . ) Then? ( ^{mathrm{A}} cdot frac{1}{2} leq I leq 1 ) в. ( 4 leq I leq 2 sqrt{30} ) ( frac{sqrt{3}}{8} leq I leq frac{sqrt{2}}{6} ) D. ( 1 leq I leq frac{2 sqrt{3}}{sqrt{2}} ) | 12 |
| 1074 | When the mean value theorem does apply? This question has multiple correct options A. Function needs to be continuous B. Function needs to be differentiable C. Function needs to be non-differentiable D. None of the above | 12 |
| 1075 | 31/4 dx To (1999 – 2 Mark: – is equal to 1+cos x (6) 2 (a) 2 (c) 1/2 (d) -1/2 | 12 |
| 1076 | Evaluate ( int frac{d x}{1+sqrt{x^{2}+2 x+2}} ) ( mathbf{A} cdot I=ln (x+1-sqrt{x^{2}+2 x+2})+ ) ( frac{2}{(x+2)+sqrt{x^{2}+2 x+2}}+C ) B ( cdot I=ln (x-2-sqrt{x^{2}-2 x-4})+ ) ( frac{2}{(x+2)+sqrt{x^{2}+2 x+2}}+C ) C ( . I=ln (x+1+sqrt{x^{2}+2 x+2})+ ) ( frac{2}{(x+2)+sqrt{x^{2}+2 x+2}}+C ) D. None of these | 12 |
| 1077 | ( int frac{cos 4 x}{sin 2 x} d x ) | 12 |
| 1078 | Evaluate ( int sqrt{1+y^{2}} cdot 2 y d y ) A ( cdot I=frac{2}{3}left(1+y^{2}right)^{3 / 2}+C ) B. ( _{I=} frac{2}{5}left(1-y^{2}right)^{3 / 2}+C ) c. ( _{I=} frac{2}{3}left(1-y^{2}right)^{3 / 2}+C ) D. None of these. | 12 |
| 1079 | The value of ( int sqrt{frac{e^{x}}{e^{x}}+1} d x ) is equal to ( mathbf{A} cdot ln left(e^{x}+sqrt{e^{2} x}-1right)-sec ^{-} 1left(e^{x}right)+c ) B ( cdot ln left(e^{x}+sqrt{e^{2} x}-1right)+sec ^{-} 1left(e^{x}right)+c ) C ( cdot ln left(e^{x}-sqrt{e^{2} x}-1right)-sec ^{-} 1left(e^{x}right)+c ) D ( cdot ln left(e^{x}+sqrt{e^{2} x}-1right)-sin ^{-} 1left(e^{-x}right)+c ) | 12 |
| 1080 | Write the formula for integration by parts. | 12 |
| 1081 | Evaluate the given integral: ( int_{0}^{1}(1- ) ( left.boldsymbol{x}^{2}right) boldsymbol{d} boldsymbol{x} ) | 12 |
| 1082 | Assertion f ( n>1 ) then Statement – 1 : ( int_{0}^{infty} frac{boldsymbol{d} boldsymbol{x}}{mathbf{1}+boldsymbol{x}^{n}}=int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{left(mathbf{1}-boldsymbol{x}^{n}right)^{1 / n}} ) Reason Statement -2: ( int_{a}^{b} f(x) d x=int_{a}^{b} f(a+ ) ( boldsymbol{b}-boldsymbol{x}) d boldsymbol{x} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1083 | Evaluate : ( intleft(1+2 x+3 x^{2}+4 x^{3}+dotsright) d x ) ( |boldsymbol{x}|<mathbf{1}) ) | 12 |
| 1084 | 18. Let f() = f 12–edt . Then the real roots of the equation x2 – f ‘(x) = 0 are (2002) (a) 1 It (b) I (c) (d) 0 and 1 | 12 |
| 1085 | ( intleft[log (1+cos x)-x tan left(frac{x}{2}right)right] d x ) A ( cdot x log (1+tan x) ) B ( cdot x log (1+sin x) ) ( mathbf{C} cdot log (1+sin x) ) D. ( x log (1+cos x) ) | 12 |
| 1086 | Write the value of ( int frac{d x}{x^{2}+16} ) | 12 |
| 1087 | ( lim _{n rightarrow infty} sum_{r=1}^{n} frac{1}{sqrt{4 n^{2}-r^{2}}}= ) A ( cdot frac{pi}{2} ) в. c. D. ( frac{pi}{5} ) | 12 |
| 1088 | ( operatorname{Let} boldsymbol{S}_{boldsymbol{n}}=frac{boldsymbol{n}}{(boldsymbol{n}+mathbf{1})(boldsymbol{n}+mathbf{2})}+ ) ( frac{n}{(n+2)(n+4)}+frac{n}{(n+3)(n+6)}+ ) ( ldots .+frac{1}{6 n}, ) then ( lim _{n rightarrow infty} S_{n} ) is This question has multiple correct options A ( cdot ln frac{3}{2} ) B. ( ln frac{9}{2} ) ( c cdot>1 ) D . < 2 | 12 |
| 1089 | ( int_{-pi / 2}^{pi / 2} frac{d x}{theta^{sin x}+1} ) is equal to ( A cdot-frac{pi}{2} ) в. ( frac{pi}{2} ) ( c cdot 0 ) D. | 12 |
| 1090 | ( int_{-2}^{0}left(x^{3}+3 x^{2}+3 x+(x+1) cos (x+right. ) 1) ( d x ) | 12 |
| 1091 | Solve : ( int frac{x+4}{x^{3}+3 x^{2}-10 x} d x ) A ( cdot frac{2}{5} ln |x|-frac{3}{7} ln |x-2|+frac{1}{35} ln |x+5|+c ) B. ( -frac{2}{5} ln |x|+frac{3}{7} ln |x-2|-frac{1}{35} ln |x+5|+c ) c. ( frac{2}{5} ln |x|+frac{3}{7} ln |x-2|+frac{1}{35} ln |x+5|+c ) D. ( -frac{2}{5} ln |x|-frac{3}{7} ln |x-2|-frac{1}{35} ln |x+5|+c ) | 12 |
| 1092 | Evaluate: ( int_{-1}^{2}left|x^{3}-xright| d x ) | 12 |
| 1093 | Evaluate: ( int frac{sin ^{6} x+cos ^{6} x}{sin ^{2} x cos ^{2} x} d x ) | 12 |
| 1094 | Evaluate the following integration w.r.t. ( boldsymbol{x} ) ( int frac{1}{(4 x+5)^{2}+1} d x ) | 12 |
| 1095 | Repeated application of integration by parts gives us the reduction formula, if the integrand is dependent on a natural number ( n ) If ( int frac{cos ^{m} x}{sin ^{n} x} d x=frac{cos ^{m-1} x}{(m-n) sin ^{n-1} x}+ ) ( A int frac{cos ^{m-2} x}{sin ^{n} x} d x+C, ) then ( A ) is equal to A ( cdot frac{m}{m+n} ) в. ( frac{m-1}{m+n} ) c. ( frac{m}{m+n-1} ) D. ( frac{m-1}{m-n} ) | 12 |
| 1096 | Solve ( int frac{x^{6}-1}{1+x^{2}} d x ) | 12 |
| 1097 | ( int frac{1}{x sqrt{x^{2}-1}} d x ) is equal to ( mathbf{A} cdot cos ^{-1} x+C ) B . ( sec ^{-1} x+C ) ( mathbf{c} cdot cot ^{-1} x+C ) D. ( tan ^{-1} x+C ) | 12 |
| 1098 | ( int_{3-alpha}^{3+alpha} f(x) d x ) equals, where ( f(3+beta)= ) ( f(3-beta), beta in R ) A. ( _{3} int_{0}^{alpha} f(x) d x ) в. ( 3 int_{0}^{3} f(x) d x ) c. ( 3 int_{alpha-3}^{alpha} f(x) d x ) D. ( 3 int_{3}^{3+alpha} f(x) d x ) | 12 |
| 1099 | Obtain as the limit of sum ( int_{log _{e}^{3}} e^{x} d x ) | 12 |
| 1100 | If ( int frac{3 x+4}{x^{3}-2 x-4} d x=log |x-2|+ ) ( boldsymbol{K} log boldsymbol{f}(boldsymbol{x})+boldsymbol{C}, ) then This question has multiple correct options A. ( K=-1 / 2 ) B . ( f(x)=x^{2}+2 x+2 ) C ( . f(x)=left|x^{2}+2 x+2right| ) D. ( K=1 / 4 ) | 12 |
| 1101 | Evaluate ( : int frac{log x}{x} d x ) | 12 |
| 1102 | Observe the following Lists List-I A: ( int_{-2}^{2} frac{1}{4+x^{2}} d x ) List-II B: ( int_{1}^{2} frac{1}{x sqrt{x^{2}-1}} d x quad ) 1) ( frac{pi}{3} ) C: ( int_{0}^{pi} cos 3 x cdot cos 2 x d x quad ) 2) 0 4) ( frac{pi}{2} ) A. A-3, B-1, C-4 B. A-3, B-1, C-2 C. A-1, B-3, C-2 D. A-4, B-1, C-2 | 12 |
| 1103 | ( int 7^{7^{7^{x}}} cdot 7^{7^{x}} cdot 7^{x} d x= ) ( ^{mathbf{A}} cdot frac{7^{7^{7^{x}}}}{(log 7)^{3}}+C ) в. ( frac{7^{7^{7}}}{(log 7)^{2}}+C ) C ( cdot 7^{7^{7^{x}}} cdot(log 7)^{3}+C ) D・ ( 7^{7^{7}} ) | 12 |
| 1104 | ( int x sqrt{frac{a^{2}-x^{2}}{a^{2}+x^{2}}} d x ) is equal to A ( cdot frac{a^{2}}{2} sin ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{4}-x^{4}}+C ) B ( cdot frac{a^{2}}{2} tan ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{4}-x^{4}}+C ) C ( cdot frac{a^{2}}{2} sin ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{3}{2} sqrt{a^{2}-x^{2}}+C ) D ( frac{a^{2}}{2} sin ^{-1}left(frac{x^{2}}{a^{2}}right)+frac{1}{2} sqrt{a^{2}-x^{2}}+C ) | 12 |
| 1105 | Find ( int frac{e^{x}(x-3)}{(x-1)^{3}} d x ) | 12 |
| 1106 | ( int frac{x^{2}}{(x sin x+cos x)^{2}} d x ) A. ( cot x+frac{x csc x}{x cos x-sin x}+c ) в. ( frac{sin x-x cos x}{x sin x+cos x}+c ) c. ( -cot x+frac{x csc x}{x cos x-sin x}+c ) D. ( cot x-frac{x csc x}{x cos x-sin x}+c ) | 12 |
| 1107 | – 8. Letf:R → R be a differentiable function havingf (2) , f(x) 41² (2)-(C). Then lingua dit quals 12005) Then lim dt equals [2005] 26 x -2° (a) 24 (6) 36 (c) 12 (d) 18 | 12 |
| 1108 | ( int frac{2 d x}{x^{2}-1} ) equals: ( mathbf{A} cdot frac{1}{2} log left(frac{x+1}{x-1}right)+C ) B ( cdot frac{1}{2} log left(frac{x-1}{x+1}right)+C ) ( mathbf{c} cdot log left(frac{x+1}{x-1}right)+C ) D ( log left(frac{x-1}{x+1}right)+C ) | 12 |
| 1109 | Evaluate ( int_{0}^{2} frac{1}{sqrt{3+2 x-x^{2}}} d x ) | 12 |
| 1110 | ( operatorname{Let} frac{mathbf{d}}{mathbf{d} x} boldsymbol{F}(boldsymbol{x})=frac{e^{sin x}}{boldsymbol{x}}, boldsymbol{x}>mathbf{0} ) ( int_{1}^{4} frac{2 e^{sin x^{2}}}{x} d x=F(k)-F(1) ) then one of the possible value of k is | 12 |
| 1111 | ( int frac{d x}{xleft(x^{7}+1right)} ) is equal to: ( ^{mathbf{A}} cdot log left(frac{x^{7}}{x^{7}+1}right) ) в. ( frac{1}{7} log left(frac{x^{7}}{x^{7}+1}right)+c ) ( ^{mathbf{c}} cdot log left(frac{x^{7}+1}{x^{7}}right)+c ) D ( cdot frac{1}{7} log left(frac{x^{7}+1}{x^{7}}right)+c ) | 12 |
| 1112 | Integrate the function ( frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}} ) | 12 |
| 1113 | Integrate: ( int frac{sin ^{-1} x}{sqrt{1-x^{2}}} d x ) | 12 |
| 1114 | The value of ( int_{0} overline{mathbf{2}} log left(frac{mathbf{4}+mathbf{3} sin boldsymbol{x}}{mathbf{4}+mathbf{3} cos boldsymbol{x}}right) boldsymbol{d} boldsymbol{x} ) is A . 2 B. ( frac{3}{4} ) ( c cdot 0 ) D. – 2 | 12 |
| 1115 | ( int frac{sec ^{2} x}{sqrt{operatorname{asec}^{2} x-operatorname{btan}^{2} x}} d x ) is (where ( c ) is integration constant This question has multiple correct options A ( cdot frac{1}{sqrt{b-a}} sin ^{-1}(tan x sqrt{frac{b-a}{a}})+c ) if ( b>a>0 ) B. ( frac{1}{sqrt{b-a}} log _{2}(tan x sqrt{b-a}+sqrt{operatorname{asec}^{2} x-b tan ^{2} x})+c ) if ( b>a>0 ) c. ( frac{1}{sqrt{a-b}} sin ^{-1}(tan x sqrt{frac{a-b}{a}})+c ) if ( a>b>0 ) D. ( frac{1}{sqrt{a-b}} log _{e}(tan x sqrt{a-b}+sqrt{operatorname{asec}^{2} x-b tan ^{2} x})+c ) if ( a>b>0 ) | 12 |
| 1116 | If ( boldsymbol{b}>boldsymbol{a} ) and ( boldsymbol{I}=int_{a}^{b} frac{boldsymbol{d} boldsymbol{x}}{sqrt{(boldsymbol{x}-boldsymbol{a})(boldsymbol{b}-boldsymbol{x})}} ) then ( I ) equals A . ( pi / 2 ) в. ( pi ) ( mathrm{c} cdot 3 pi / 2 ) D . 2 ( pi ) | 12 |
| 1117 | Solve ( frac{1}{2} int frac{(-4+2 x)}{sqrt{5-4 x+x^{2}}} ) | 12 |
| 1118 | If ( frac{mathbf{3} boldsymbol{x}+mathbf{2}}{(boldsymbol{x}+mathbf{1})left(mathbf{2} boldsymbol{x}^{2}+mathbf{3}right)}=frac{boldsymbol{A}}{(boldsymbol{x}+mathbf{1})}+ ) ( frac{B x+C}{left(2 x^{2}+3right)} ) then ( A+C-B= ) A. B. 2 ( c cdot 3 ) ( D ) | 12 |
| 1119 | Evaluate ( int x^{2} log x d x ) A. ( frac{x^{2}}{2} log x-frac{1}{9} x^{2}+c ) B ( cdot frac{x^{3}}{3} log x-frac{1}{9} x^{2}+c ) c. ( frac{x^{3}}{3} log x-frac{1}{9} x^{3}+c ) D. ( frac{x^{3}}{3} log x+frac{1}{9} x^{3}+c ) | 12 |
| 1120 | If ( boldsymbol{I}_{n}=int_{0}^{frac{pi}{4}} tan ^{n} x d x ) then ( frac{1}{I_{2}+I_{4}}, frac{1}{I_{3}+I_{5}}, frac{1}{I_{4}+I_{6}} ) are in? A . ( A . P ) в. ( H . P ) c. ( G . P ) D. None of these | 12 |
| 1121 | ( f(x-2)(x-3)^{3}=frac{A}{x-2}+frac{B}{x-3}+ ) ( frac{C}{(x-3)^{2}}+frac{D}{(x-3)^{3}} ) then ( B= ) ( A ) B. c. ( frac{1}{25} ) ( D ) | 12 |
| 1122 | Evaluate ( int frac{d x}{sqrt{x+1}-sqrt{x}} ) | 12 |
| 1123 | In (1+ 8x dx 5x + x) 9. Find the indefinite integral dl 3.2460 | 12 |
| 1124 | ( n stackrel{L t}{rightarrow} inftyleft{frac{1}{n+1}+frac{1}{n+2}+ldots+frac{1}{6 n}right}= ) ( A cdot log 2 ) B. ( log 3 ) ( c cdot log 5 ) ( D cdot log 6 ) | 12 |
| 1125 | Evaluate ( int e^{x}left(log (x)+frac{1}{x^{2}}right) d x ) A. ( e^{x}left(log x+frac{1}{x^{2}}right) ) B. ( quad e^{x}left(log x+frac{1}{x}right) ) c. ( quad e^{x}left(log x-frac{1}{x^{2}}right) ) D. ( quad e^{x}left(log x-frac{1}{x}right) ) | 12 |
| 1126 | ( int frac{3+4 sin x+2 cos x}{3+2 sin x+cos x} d x ) A ( cdot 2 x+3 tan ^{-1}left(tan frac{x}{2}+1right)+c ) B . ( 2 x-3 tan ^{-1}left(tan frac{x}{2}+1right)+c ) c. ( 2 x-6 tan ^{-1}left(tan frac{x}{2}+1right)+c ) D・ ( x-3 tan ^{-1}left(tan frac{x}{2}+1right)+c ) | 12 |
| 1127 | Suppose we define definite integral using the formula ( int_{a}^{b} f(x) d x= ) ( frac{b-a}{2}{f(a)+f(b)} . ) For more accurate result, we have ( int_{a}^{b} f(x) d x= ) ( frac{b-a}{4}{f(a)+f(b)+2 f(c)}, ) when ( c=frac{a+b}{2} cdot ) Also, let ( F(c)= ) ( frac{c-a}{2}{f(a)+f(c)}+ ) ( frac{b-c}{2}{f(b)+f(c)}, ) when ( c epsilon(a, b) ) (i) ( int_{0}^{pi / 2} sin x d x ) equals A ( cdot frac{pi}{8}(1+sqrt{2}) ) в. ( frac{pi}{4}(1+sqrt{2}) ) c. ( frac{pi}{8 sqrt{2}} ) D. ( frac{pi}{4 sqrt{2}} ) | 12 |
| 1128 | Find: ( int x^{2} cdot log x d x ) | 12 |
| 1129 | Evaluate ( int_{0}^{2}left(x^{2}+2 x+1right) d x ) | 12 |
| 1130 | Evaluate ( : int_{-pi / 4}^{pi / 4} x^{5} cos ^{2} x d x ) | 12 |
| 1131 | ( int_{0}^{pi / 4} sec ^{2} x d x ) | 12 |
| 1132 | The value of the definite integral ( int_{0} sqrt{ln left(frac{pi}{2}right)} cos left(e^{x^{2}}right) 2 x e^{x^{2}} d x ) is: ( A cdot 1 ) B. ( 1+(sin 1) ) c. ( 1-(sin 1) ) D. ( (sin 1)-1 ) | 12 |
| 1133 | ( int frac{sin x+4 sin 3 x+6 sin 5 x+3 sin 7 x}{sin 2 x+3 sin 4 x+3 sin 6 x} ) equals. ( mathbf{A} cdot-2 sin x+c ) B. ( 2 sin x+c ) c. ( 2 cos x+c ) D. ( -2 cos x+c ) | 12 |
| 1134 | Evaluate ( : int frac{1}{sin x-sin 2 x} d x ) | 12 |
| 1135 | ( int_{infty}^{a} frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{x}^{4} sqrt{left(boldsymbol{a}^{2}+boldsymbol{x}^{2}right)}}=frac{boldsymbol{k}-sqrt{boldsymbol{k}}}{mathbf{3} boldsymbol{a}^{4}} ) What is ( k ? ) | 12 |
| 1136 | ntegrate the function ( frac{1}{sqrt{9 x^{2}+6 x+5}} ) | 12 |
| 1137 | 11. If y = x², then yox will be: (d) O | 12 |
| 1138 | Find the values of ( c ) that satisfy the MVT for integrals on ( left[frac{3 pi}{4}, piright] ) ( boldsymbol{f}(boldsymbol{x})=cos (boldsymbol{2} boldsymbol{x}-boldsymbol{pi}) ) A ( cdot c=frac{5 pi}{2}-frac{1}{2} cos ^{-1}left(-frac{2}{pi}right) ) в. ( c=pi-frac{1}{2} cos ^{-1}left(-frac{2}{pi}right) ) ( c cdot c=frac{pi}{2}+frac{1}{2} sin ^{-1}left(-frac{2}{pi}right) ) D ( c=frac{pi}{2}+frac{1}{2} sin ^{-1}left(frac{2}{pi}right) ) | 12 |
| 1139 | Evaluate ( int frac{x^{2}}{(x sin x+cos x)^{2}} d x ) A. ( frac{sin x-x cos x}{x sin x+cos x}+c ) в. ( frac{cos x-x sin x}{x sin x+cos x}+c ) c. ( frac{cos x-x sin x}{x sin x-cos x}+c ) D. ( frac{sin x+x cos x}{x sin x+cos x}+c ) | 12 |
| 1140 | 13a + 4x² 44. Find the value of J -1/3 2- dx | 12 |
| 1141 | Let ( boldsymbol{f}(boldsymbol{x}) ) denotes the fractional part of ( mathbf{a} ) real number ( x ). Then the value of ( int_{0}^{sqrt{3}} fleft(x^{2}right) d x ) A ( cdot 2 sqrt{3}-sqrt{2}-1 ) 1 1 в. ( 0(z e r o) ) c. ( sqrt{2}-sqrt{3}+1 ) D. ( sqrt{3}-sqrt{2}+1 ) | 12 |
| 1142 | Evaluate the following integral: ( int frac{x^{3}-3 x^{2}+5 x-7+x^{2} a^{x}}{2 x^{2}} d x ) | 12 |
| 1143 | 35. The value of 8log(1+x) dx is 1 + x2 (a) log2 (b) log2 (c) log 2 (d) a log 2 | 12 |
| 1144 | ( f int_{0}^{2 pi} log (1+sin x) d x=k pi log frac{1}{2} ) then find the value of ( k ) | 12 |
| 1145 | ( int_{-1}^{1} x(1-x)(1+x) d x ) is equal to ( A cdot frac{1}{3} ) B. ( frac{2}{3} ) c. 1 D. – E . | 12 |
| 1146 | ( int frac{boldsymbol{x}+boldsymbol{2}}{mathbf{2} boldsymbol{x}^{2}-mathbf{7} boldsymbol{x}+mathbf{3}} boldsymbol{d} boldsymbol{x}= ) A. ( log left|frac{x-3}{2 x-1}right|+c ) в. ( log left|frac{x-3}{sqrt{2 x-1}}right|+c ) c. ( frac{1}{2} log left|frac{x-3}{2 x-1}right|+c ) D. ( frac{1}{2} log left|frac{x-3}{sqrt{2 x-1}}right|+c ) | 12 |
| 1147 | ( int sqrt{frac{cos x-cos ^{3} x}{1-cos ^{3} x}} d x ) is equal to A ( cdot frac{2}{3} sin ^{-1}left(cos ^{3 / 2} xright)+C ) B ( cdot frac{3}{2} sin ^{-1}left(cos ^{3 / 2} xright)+C ) C ( cdot frac{2}{3} cos ^{-1}left(cos ^{3 / 2} xright)+C ) D. none of these | 12 |
| 1148 | The integral ( int_{0}^{a} frac{g(x)}{f(x)+f(a-x)} d x ) vanishes, if A ( cdot g(x) ) is odd B. ( f(x)=f(a-x) ) c. ( g(x)=-g(a-x) ) D. ( f(a-x)=-g(x) ) | 12 |
| 1149 | Integrate: ( 2 x^{2} e^{x^{2}} ) | 12 |
| 1150 | Evaluate the given integral. ( int sin ^{-1}left(3 x-4 x^{3}right) d x ) | 12 |
| 1151 | ( sqrt{x^{2}+2 x+5} d x ) is equal to ( mathbf{A} cdot(x+1) sqrt{x^{2}+2 x+5}+frac{1}{2} log |x+1+sqrt{x^{2}+2 x+5}|+C ) В . ( (x+1) sqrt{x^{2}+2 x+5}+2 log |x+1+sqrt{x^{2}+2 x+5}|+C ) c. ( (x+1) sqrt{x^{2}+2 x+5-2} log |x+1+sqrt{x^{2}+2 x+5}|+C ) D ( cdot frac{1}{2}(x+1) sqrt{x^{2}+2 x+5}+2 log |x+1+sqrt{x^{2}+2 x+5}|+ ) | 12 |
| 1152 | Solve : ( int x sqrt{1+2 x^{2}} d x ) | 12 |
| 1153 | Evaluate ( : int frac{e^{2 x}-1}{e^{2 x}+1} d x ) | 12 |
| 1154 | 1 (x-1) et 4. Evaluate : (x+13 ax | 12 |
| 1155 | Solve ( int_{1}^{-1} frac{d}{d x} tan ^{-1}left(frac{1}{x}right) d x ) | 12 |
| 1156 | Evaluate: ( int frac{sec ^{8} x}{operatorname{cosec} x} d x ) | 12 |
| 1157 | Evaluate ( int_{0}^{pi / 2} frac{cos x}{1+sin ^{2} x} d x ) | 12 |
| 1158 | Write the value of ( int boldsymbol{X} boldsymbol{a}^{boldsymbol{x}^{2}+1} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1159 | ( int frac{dleft(x^{2}+1right)}{sqrt{x^{2}+2}} ) is equal to A ( cdot 2 sqrt{x^{2}+2}+k ) B . ( sqrt{x^{2}+2}+k ) ( left(frac{1}{x^{2}+2}right)^{frac{3}{2}}+k ) D. none of these | 12 |
| 1160 | 20. Let f(x),x 20, be a non-negative continuous function, and let F(x) = f(t)dt,x20. If for some c>0,f(x) ScF(x) for all x20, then show that f(x)=0 for all x > 0. (2001 – 5 Marks) | 12 |
| 1161 | If ( int_{1}^{2} e^{x^{2}} d x=a, ) then ( int_{e}^{e^{4}} sqrt{ln x} d x ) is equal to A ( cdot 2 e^{4}-2 e-a ) B ( cdot 2 e^{4}-e-a ) ( mathbf{c} cdot 2 e^{4}-e-2 a ) D. ( e^{4}-e-a ) | 12 |
| 1162 | The value of ( lim _{n rightarrow infty} frac{(n !) frac{1}{n}}{n} ) is? ( mathbf{A} cdot mathbf{1} ) в. ( frac{1}{e^{2}} ) c. ( frac{1}{2 e} ) D. | 12 |
| 1163 | Prove that: ( int frac{x^{2} d x}{(x sin x+cos x)^{2}} ) | 12 |
| 1164 | Evaluate ( int(1-x) sqrt{x} . d x ) | 12 |
| 1165 | Let mean value of ( boldsymbol{f}(boldsymbol{x})=frac{1}{boldsymbol{x}+boldsymbol{c}} ) over interval (0,2) is ( frac{1}{2} ell n 3 ) then positive values of ( c ) is A . 12 в. 1 ( c cdot 2 ) ( D cdot frac{3}{3} ) | 12 |
| 1166 | Prove that: ( int_{0}^{pi} frac{x d x}{1+sin x}=pi ) | 12 |
| 1167 | Solve: ( int frac{1}{x-x^{3}} d x ) | 12 |
| 1168 | ( int frac{x d x}{left(x^{2}+a^{2}right)left(x^{2}+b^{2}right)} ) | 12 |
| 1169 | Solve ( int frac{x^{5}}{x^{2}+1} d x ) A. ( frac{x^{4}}{4}+frac{x^{2}}{2}+tan ^{-1} x+c ) B. ( frac{x^{4}}{4}-frac{x^{2}}{2}+frac{1}{2} log left(x^{2}+1right)+c ) C ( frac{x^{4}}{4}-frac{x^{3}}{3}+tan ^{-1} x+c ) D. ( frac{x^{4}}{4}+frac{x^{2}}{2}+frac{1}{2} log left(x^{2}+1right)+c ) | 12 |
| 1170 | ( int e^{x}left[log (cosh x)-operatorname{sech}^{2} xright] d x= ) A ( cdot e^{x}(log cosh x-tanh x)+c ) B. ( e^{x} log cosh x+c ) c. ( -e^{x} tanh x+c ) D. ( e^{x}(log cosh x+tanh x)+c ) | 12 |
| 1171 | If ( f(x) ) be a quadratic polynomial such that ( boldsymbol{f}(mathbf{0})=mathbf{2}, boldsymbol{f}^{prime}(mathbf{0})=-mathbf{3} ) and ( boldsymbol{f}^{prime prime}(mathbf{0})=mathbf{4} ) then ( int_{-1}^{1} f(x) d x ) is equal to A . -3 в. ( 16 / 3 ) c. 0 D. none of these | 12 |
| 1172 | If ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{-3} ) then ( boldsymbol{y}= ) A ( cdot frac{-1}{2 x^{2}}+c ) B ( cdot frac{-x^{-4}}{4}+c ) c. ( frac{2}{x^{2}}+c ) D. ( frac{x^{-2}}{2}+c ) | 12 |
| 1173 | If /(m, n) = (1+t)” dt, then the expression for l(m, n) in (2003) terms of l(m +1, n-1) is 21 n -“(m +1, n-1) m +1 m +1 п – -1(m +1, n-1) m+1 ann -+-^— 1(m+1, n-1) +1 m +1 m т. “-1(m +1, n-1) n+1 | 12 |
| 1174 | ( int e^{x}left[log cos x+sec ^{2} xright] d x= ) A ( cdot e^{x}left[log cos x+sec ^{2} xright]+c ) B . ( e^{x}[log cos x+tan x]+c ) c. ( e^{x}(cos x)+c ) D ( cdot e^{x}[log (tan x)]+c ) | 12 |
| 1175 | Solve: ( int_{1}^{2} frac{2}{x} d x ) | 12 |
| 1176 | ( int_{0}^{infty} fleft(x+frac{1}{x}right) frac{ln x}{x} d x ) A. is equal to zero B. is equal to one ( mathrm{c} cdot_{text {is equal to }} frac{1}{2} ) D. can not be evaluated | 12 |
| 1177 | Solve: ( int frac{d x}{sqrt{4-x^{2}}} ) | 12 |
| 1178 | The value of ( lim _{n rightarrow infty}left[frac{sqrt{boldsymbol{n}+mathbf{1}}+sqrt{boldsymbol{n}+mathbf{2}}+ldots+sqrt{boldsymbol{n}+boldsymbol{r}}}{boldsymbol{n} sqrt{boldsymbol{n}}}right. ) is A ( cdot frac{2(2 sqrt{2}-1)}{3} ) B. ( frac{(2 sqrt{2}-1)}{3} ) c. ( frac{(2 sqrt{2}+1)}{3} ) D. ( frac{(sqrt{2}+1)}{3} ) | 12 |
| 1179 | Solve: ( int frac{cos x}{6+4 sin x-cos ^{2} x} d x ) | 12 |
| 1180 | If ( I=int_{0}^{pi}left(pi x-x^{2}right)^{100} sin 2 x d x, ) then value of ( I ) is? A . ( pi^{100} ) B. ( frac{1}{2}left(pi^{100}-pi^{97}right) ) c. ( frac{1}{2}left(pi^{100}+pi^{97}right) ) D. | 12 |
| 1181 | ( int frac{1}{left(x^{6}-1right)} d x ) A ( cdot 1 / 2left(frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-frac{1}{sqrt{3}} arctan frac{2 x+1}{sqrt{3}}-frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}-right. ) ( left.quad frac{1}{sqrt{3}} arctan frac{2 x-1}{sqrt{3}}right)+mathrm{k} ) B ( cdot ) [ begin{array}{l}text { C } mid / 2left(frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}-frac{1}{sqrt{3}} arctan frac{2 x-1}{sqrt{3}}-frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-right. \ left.frac{1}{sqrt{3}} arctan frac{2 x+1}{sqrt{3}}right)+mathrm{k}end{array} ] C ( cdot 1 / 2left(frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-frac{1}{sqrt{3}} operatorname{arccot} frac{2 x+1}{sqrt{3}}+frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}+right. ) ( left.frac{1}{sqrt{3}} operatorname{arccot} frac{2 x-1}{sqrt{3}}right)+mathrm{k} ) D ( cdot 1 / 2left(frac{1}{6} ln frac{(x-1)^{2}}{x^{2}+x+1}-frac{1}{sqrt{3}} operatorname{arccot} frac{2 x+1}{sqrt{3}}-frac{1}{6} ln frac{(x+1)^{2}}{x^{2}-x+1}-right. ) ( left.frac{1}{sqrt{3}} operatorname{arccot} frac{2 x-1}{sqrt{3}}right)+mathrm{k} ) | 12 |
| 1182 | If ( int f(x) d x=f(x), ) then ( int[j(x)]^{2} d x ) is A ( cdot frac{1}{2}[f(x)]^{2} ) B . ( [f(x)]^{3} ) c. ( frac{[f(x)]^{3}}{3} ) D cdot ( [f(x)]^{2} ) | 12 |
| 1183 | Solve: ( int frac{x^{2}+1}{x^{2}-4 x+6} d x ) | 12 |
| 1184 | Consider the integral ( boldsymbol{I}= ) ( int_{0}^{pi} ln (sin x) d x . ) What is ( int_{0}^{frac{pi}{2}} ln ) ( (sin x) d x ) equal to? A . ( 4 I ) B. 2I ( I ) c. ( I ) D. ( frac{I}{2} ) | 12 |
| 1185 | Solve ( int boldsymbol{a}^{boldsymbol{m} boldsymbol{x}} boldsymbol{b}^{boldsymbol{n} boldsymbol{x}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1186 | ( int_{0}^{2 x} sqrt{1+sin x} d x ) A ( cdot sin frac{x}{2}-cos frac{x}{2}-frac{pi}{2}+C ) B. ( sin frac{x}{2}-cos frac{x}{2}+frac{pi}{2}+C ) c. ( 2 sin x-2 cos x+2 ) D. None of these | 12 |
| 1187 | Evaluate the following definite integrals ( int_{0}^{pi / 2} cos ^{2} x d x ) | 12 |
| 1188 | ( int|x+y| d x, ) where ( frac{d y}{d x}=0 ) is given by A . 0 в. ( frac{(x+y)^{2}}{2}+c ) c. ( -frac{(x+y)^{2}}{2}+c ) D. ( frac{(x+y)|x+y|}{2}+c ) | 12 |
| 1189 | ( int frac{4 sec ^{2} x tan x}{sec ^{2} x+tan ^{2} x} d x= ) A ( cdot 2 log left(sec ^{2} x+tan ^{2} xright)+c ) B ( cdot log left(2 x+tan ^{2} xright)+c ) c. ( 2 tan ^{2} x+c ) ( mathbf{D} cdot log left(sec ^{2} x+tan ^{2} xright)+c ) | 12 |
| 1190 | Solve ( int frac{cos x}{1+cos x} d x ) | 12 |
| 1191 | ( int_{0}^{5} x^{3}left(25-x^{2}right)^{7 / 2} d x ) | 12 |
| 1192 | ( int_{0}^{infty} frac{log x}{1+x^{2}} d x ) | 12 |
| 1193 | Evaluate the following definite integral: ( int_{0}^{2} frac{1}{4+x-x^{2}} d x ) | 12 |
| 1194 | Evaluate the following integral as limit of sum: ( int_{1}^{4}left(x^{2}-xright) d x ) | 12 |
| 1195 | Evaluate ( int frac{d x}{x+4-x^{2}} ) | 12 |
| 1196 | If ( I=int_{alpha}^{beta}left[log log x+frac{1}{(log x)^{2}}right] d x ) then ( boldsymbol{I} ) equals A. ( alpha log log alpha-beta log log beta ) B. ( frac{1}{alpha}-frac{1}{beta}+log log alpha-log log beta ) c. ( frac{beta-alpha}{alpha beta}+alpha log log alpha-beta log log beta ) D. none of these | 12 |
| 1197 | Evaluate integral of ( int frac{d x}{sqrt[4]{1+x^{4}}}, ) the ans is ( A ) [ -frac{1}{4}left[log frac{sqrt[4]{1+1 / x^{4}}-1}{sqrt[4]{left(1+1 / x^{4}right)}+1}-2 tan ^{-1}left(1+frac{1}{x^{4}}right)^{frac{1}{4}}right] ] B. [ -frac{1}{4}left[log frac{sqrt[4]{1+1 / x^{4}}-1}{sqrt[4]{left(1+1 / x^{4}right)}+1}+2 tan ^{-1}left(1+frac{1}{x^{4}}right)^{frac{1}{4}}right]+C ] ( c ) [ frac{1}{4}left[log frac{sqrt[4]{1+1 / x^{4}}-1}{sqrt[4]{left(1+1 / x^{4}right)}+1}+2 tan ^{-1}left(1+frac{1}{x^{4}}right)^{frac{1}{4}}right] ] ( D ) [ frac{1}{4}left[log frac{sqrt[4]{1+1 / x^{4}}-1}{sqrt[4]{left(1+1 / x^{4}right)}+1}-2 tan ^{-1}left(1+frac{1}{x^{4}}right)^{frac{1}{4}}right]+C ] | 12 |
| 1198 | c os + sin 10. Find the indefinite integral ſcos 20 ln cos 0 – sino) indefinite integral co | 12 |
| 1199 | Evaluate the following integral ( int frac{e^{x}+1}{e^{x}+x} d x ) | 12 |
| 1200 | Integrate ( int_{1}^{2} x^{2} d x ) | 12 |
| 1201 | Evaluate ( : int frac{1+log x}{x(2+log x)(3+log x)} d x ) | 12 |
| 1202 | Evaluate: ( int frac{d x}{left(x^{2}+1right)left(x^{2}+4right)} ) | 12 |
| 1203 | Evaluate the given integral. ( int frac{x^{2}-1}{x^{4}+x^{2}+1} d x ) | 12 |
| 1204 | Evaluate the integral ( int_{1}^{2}left(frac{1}{x}-frac{1}{2 x^{2}}right) e^{2 x} d x ) using substitution. | 12 |
| 1205 | Evaluate the following integral: ( int_{0}^{3} x^{2} d x ) | 12 |
| 1206 | I : Number of partial fractions of ( frac{x^{3}+x^{2}+1}{x^{4}+x^{2}+1} ) is 4 II : Number of partial fractions of ( frac{3 x+5}{(x-1)^{2}left(x^{2}+1right)^{3}} ) is 5 Which of the above statement is true. A. onlyı B. Only II c. Both I and II D. Neither I nor II | 12 |
| 1207 | & Evaluate Evaluate (Vtan x + Vcot =)dx | 12 |
| 1208 | Evaluate ( int_{0}^{2 / 3} frac{d x}{left(4+9 x^{2}right)} ) | 12 |
| 1209 | Evaluate : ( int 2^{x} d x ) | 12 |
| 1210 | ( frac{boldsymbol{x}^{2}-boldsymbol{x}-mathbf{1}}{boldsymbol{x}^{3}-mathbf{8}}=frac{boldsymbol{A}}{boldsymbol{x}-mathbf{2}}+frac{boldsymbol{B} boldsymbol{x}+boldsymbol{C}}{boldsymbol{x}^{2}+mathbf{2} boldsymbol{x}+mathbf{4}} Rightarrow ) ( boldsymbol{A}+boldsymbol{B}= ) ( A cdot 0 ) B. ( c .-1 ) ( D ) | 12 |
| 1211 | The value of ( lim _{n rightarrow infty} frac{1}{(n+1)}+frac{1}{(n+2)}+frac{1}{(n+3)}+dots ) ( ? ) A ( cdot log 4 ) в. ( log 2 ) ( c cdot log 3 ) D. ( log 5 ) | 12 |
| 1212 | ( int_{0}^{2} x sqrt{x+2}left(text { Put } x+2=t^{2}right) ) | 12 |
| 1213 | Prove that ( int_{0}^{pi / 2} frac{d x}{1+tan x}=frac{pi}{4} ) | 12 |
| 1214 | Solve ( int_{2}^{-13} frac{d x}{sqrt[5]{(3-x)^{4}}} ) A ( cdot-5(sqrt[5]{16}-1) ) B ( cdot 5(sqrt[3]{16}-1) ) ( mathbf{c} cdot-5(sqrt[5]{16}+1) ) D. None of these | 12 |
| 1215 | Prove that ( int sqrt{left(a^{2}-x^{2}right)} d x= ) ( frac{x sqrt{left(a^{2}-x^{2}right)}}{2}+frac{a^{2}}{2} sin ^{-1} frac{x}{a} ) | 12 |
| 1216 | Find ( int frac{x^{4}+1}{xleft(x^{2}+1right)^{2}} ) | 12 |
| 1217 | Suppose a continuous function ( boldsymbol{f} ) ( [0, infty) rightarrow R ) satisfies ( f(x)= ) ( 2 int_{0}^{x} t f(t) d t+1 ) for all ( x geq 0 ) Then ( boldsymbol{f}(mathbf{1}) ) equals ( A ) B ( cdot e^{2} ) ( c cdot e^{4} ) D. ( e^{6} ) | 12 |
| 1218 | Let ( p(x) ) be the fifth degree polynomial such that ( p(x)+1 ) is divisible by ( (x-1) ) and ( p(x)-1 ) is divisible by ( (x+1) . ) Then find the value of ( int_{-10}^{10} p(x) d x ) | 12 |
| 1219 | If ( boldsymbol{I}=int log (sqrt{1-boldsymbol{x}}+sqrt{1+boldsymbol{x}}) boldsymbol{d} boldsymbol{x} ) then I is equal to A ( cdot x log (sqrt{1-x}+sqrt{1+x})+frac{1}{2} x+C ) B・ ( _{x log (sqrt{1-x}+sqrt{1+x})}+frac{1}{2} sin ^{-1} x+C ) c. ( _{x log (sqrt{1-x}+sqrt{1+x})}+frac{1}{2} sin ^{-1} x-frac{1}{2} x+C ) D. ( x log (sqrt{1-x}+sqrt{1+x})+frac{1}{2} sin ^{-1} x+frac{1}{2} x+C ) | 12 |
| 1220 | Evaluate the integral ( int_{2}^{3} frac{sqrt{boldsymbol{x}}}{sqrt{mathbf{5}-boldsymbol{x}}+sqrt{boldsymbol{x}}} boldsymbol{d} boldsymbol{x} ) A ( cdot 1 / 2 ) B . ( 3 / 2 ) ( c cdot 5 / 2 ) D. 0 | 12 |
| 1221 | If ( boldsymbol{I}=int_{boldsymbol{pi} / mathbf{6}}^{boldsymbol{pi} / mathbf{3}} frac{boldsymbol{d} boldsymbol{x}}{mathbf{1}+sqrt{tan boldsymbol{x}}} ) then ( boldsymbol{I} ) equals A ( cdot frac{pi}{12} ) в. ( frac{pi}{6} ) ( c cdot frac{pi}{4} ) D. ( frac{pi}{3} ) | 12 |
| 1222 | Evaluate the following functions w.r.t. ( intleft(3 x^{2}-5right)^{2} d x ) | 12 |
| 1223 | Evaluate ( int frac{x^{2}}{xleft(1+x^{2}right)} d x ) | 12 |
| 1224 | ( int x^{2}left(1-frac{1}{x^{2}}right) d x ) | 12 |
| 1225 | ( int_{0}^{pi / 2} frac{cos 2 x}{(sin x+cos x)^{2}} d x=dots dots ) ( ^{A} cdot frac{pi}{4} ) в. ( frac{pi}{2} ) ( c cdot 0 ) ( D cdot-frac{pi}{4} ) | 12 |
| 1226 | Evaluate: ( int frac{d x}{1-tan x} ) | 12 |
| 1227 | Evaluate ( int frac{2 cos x-3 sin x}{6 cos x+4 sin x} d x ) | 12 |
| 1228 | ( int frac{csc ^{2} x-2005}{cos ^{2005} x} d x ) is equal to A. ( frac{cot x}{(cos x)^{2005}}+C ) B. ( frac{tan x}{(cos x)^{2005}}+C ) c. ( frac{-tan x}{(cos x)^{2005}}+C ) D. None of these | 12 |
| 1229 | ( L lim _{n rightarrow infty} t frac{1}{n}left[frac{1}{n+1}+frac{2}{n+2}+ldots+frac{3 n}{4 n}right] ) A. ( 3-ln 4 ) B. ( 3+ln 4 ) c. ( 3 ln 4 ) D. None of these | 12 |
| 1230 | Find the following integrals: i) ( int frac{x^{3}-1}{x^{2}} d x ) ii) ( intleft(x^{frac{2}{3}}+1right) d x ) iii) ( intleft(x^{frac{3}{2}}+2 e^{x}-frac{1}{x}right) d x ) | 12 |
| 1231 | ( int x^{2} e^{x^{3}} d x ) equals A ( cdot frac{1}{3} e^{x^{3}}+C ) B ( cdot frac{1}{3} e^{x^{2}}+C ) c. ( frac{1}{2} e^{x^{3}}+C ) D. ( frac{1}{2} e^{x^{2}}+C ) | 12 |
| 1232 | Evaluate: ( int[sin (log x)+cos (log x)] d x ) | 12 |
| 1233 | The integral ( int_{frac{pi}{4}}^{frac{3 pi}{4}} frac{d x}{1+cos x} ) A .2 B. 4 ( c cdot-1 ) D. – | 12 |
| 1234 | ( int frac{1}{xleft(x^{n}+1right)} d x ) | 12 |
| 1235 | Evaluate: ( int frac{boldsymbol{d x}}{sin ^{2} x+5 sin x cos x+2} ) | 12 |
| 1236 | ( int_{pi / 6}^{pi / 4} frac{d x}{sin 2 x} ) is equal to A ( cdot frac{1}{2} log (-1) ) B ( cdot log (-1) ) ( mathbf{c} cdot log 3 ) D ( cdot frac{1}{2} log sqrt{3} ) | 12 |
| 1237 | Solve ( int frac{sqrt{1-x^{2}}+sqrt{1+x^{2}}}{sqrt{1-x^{4}}} d x ) ( mathbf{A} cdot I=log |x-sqrt{1+x^{2}}|+sin ^{-1} x+c ) B . ( I=log |x+sqrt{1+x^{2}}|+sin ^{-1} x+c ) ( mathbf{C} cdot I=log |x+sqrt{1+x^{2}}|-sin ^{-1} x+c ) D. None of these | 12 |
| 1238 | ( int(1-x) sqrt{x} d x ) | 12 |
| 1239 | If a continuous function ( f ) satisfies ( int_{0}^{f(x)} t^{2} d t=x^{2}(1+x) ) for all ( x geq 0 ) then ( f(2) ) is equal to A . 12 B. ( sqrt[3]{36} ) ( c .3 ) D. ( sqrt[3]{42} ) | 12 |
| 1240 | Evaluate the integral ( int_{frac{a}{2}}^{a} frac{1}{sqrt{a^{2}-x^{2}}} d x ) A ( cdot frac{pi}{2} ) в. ( pi a ) c ( . pi-1 ) D. | 12 |
| 1241 | 10. Ifg(x)= | cos* t dt, then g(x+1) equals (1997 – 2 Marks) (a) g(x) + g(1) (b) g(x)-g(1) (c) g(x)g(1) g(T) (d) g(x) | 12 |
| 1242 | ( int frac{d x}{(x-b) sqrt{(x-a)(b-x)}}= ) A ( -frac{(b-a)}{2} sqrt{frac{b-x}{x-a}}+c ) В. ( -(b-a) sqrt{(b-x)(x-a)}+c ) c. ( -frac{2}{(b-a)} sqrt{frac{x-a}{b-x}}+c ) D. ( (b-a) sqrt{(x-b)(x-a)}+c ) | 12 |
| 1243 | Evaluate ( int frac{sin x+cos x}{(sin x-cos x)^{2}} d x ) A ( cdot frac{1}{sin x-cos x}+C ) в. ( frac{-1}{sin x-cos x}+C ) c. ( frac{-1}{sin x+cos x}+C ) D. None of these | 12 |
| 1244 | integrate: [ int tan ^{-1} x d x ] | 12 |
| 1245 | ( sqrt{x} e^{sqrt{x}} d x ) is equal to: A ( cdot 2 sqrt{x}-e^{sqrt{x}}-4 sqrt{x e^{sqrt{x}}}+c ) B ( cdot(2 x-4 sqrt{x}+4) e^{sqrt{x}}+c ) c. ( (2 x+4 sqrt{x}+4) e^{sqrt{x}}+c ) D. ( (1-4 sqrt{x}) e^{sqrt{x}}+c ) | 12 |
| 1246 | ( int frac{d x}{x^{2}+2 x+2}=f(x)+c Longrightarrow f(x)= ) A ( cdot tan ^{-1}(x+1) ) B. ( 2 tan ^{-1}(x+1) ) c. ( -tan ^{-1}(x+1) ) D. ( 3 tan ^{-1}(x+1) ) | 12 |
| 1247 | Evaluate ( : int frac{x-3}{(x-1)^{3}} e^{x} d x ) | 12 |
| 1248 | Evaluate: ( int frac{cos ^{3} x}{sin ^{2} x+sin x} ) | 12 |
| 1249 | r/2 dx 7. (1993 – 1 Marks) is The value of (a) o Ö 1+tan (6) 1 (c) r12 (d) a 14 | 12 |
| 1250 | If ( I_{n}=int_{0}^{infty} e^{-x} x^{n-1} d x, ) then ( int_{0}^{infty} e^{-lambda x} x^{n-1} d x ) is equal to? A ( . lambda I_{n} ) B. ( frac{1}{lambda} I ) c. ( frac{I_{mathrm{n}}}{lambda^{mathrm{n}}} ) D. ( lambda^{n} I_{n} ) | 12 |
| 1251 | If ( int frac{d x}{x^{2}+a x+1}=f(g(x))+c, ) then This question has multiple correct options A ( cdot f(x) ) is inverse trigonometric function for ( |a|>2 ) B . ( f(x) ) is logarithmic function for ( |a|2 ) D ( cdot f(x) ) is logarithmic function for ( |a|>2 ) | 12 |
| 1252 | Find ( : int frac{(2 x-5) e^{2 x}}{(2 x-3)^{3}} d x ) | 12 |
| 1253 | Evaluate: ( int_{0}^{2} frac{e^{x}}{e^{2 x}+1} d x ) | 12 |
| 1254 | Evaluate ( : int frac{sec x}{1+operatorname{cosec} x} d x ) | 12 |
| 1255 | Solve ( int frac{x}{sqrt{x+4}} d x ) | 12 |
| 1256 | Evaluate: ( int frac{2 x}{left(x^{2}+4right)} d x ) | 12 |
| 1257 | Integrate ( int frac{1}{sqrt{3-x^{2}}} d x ) | 12 |
| 1258 | Evaluate the following integrals. ( int frac{d x}{sqrt{2 x-3 x^{2}+1}} ) | 12 |
| 1259 | Evaluate the following definite integrals as limit of sums. ( int_{a}^{b} x d x ) A ( cdot b^{2}+a^{2} ) B. ( frac{b^{2}-a^{2}}{2} ) c. ( a^{2}-b^{2} ) D. ( a^{2}+b^{2} ) | 12 |
| 1260 | If ( int_{a}^{b} f(t) g o h(t)= ) ( int_{a}^{b} f o h(t) g(t) d(t), ) where ( f, g, h, ) are non negative continuous functions on ( [a, b] ) then possible choice of ( h(t) ) is This question has multiple correct options ( mathbf{A} cdot t ) B. ( a-b-t ) c. ( a+b-t ) D. ( b-t ) | 12 |
| 1261 | ( int frac{x sin ^{-1} x}{sqrt{1-x^{2}}} d x ) is equal to A. ( x-sqrt{1-x^{2}} sin ^{-1} x+c ) B. ( x+sqrt{1-x^{2}} sin ^{-1} x+c ) c. ( x+sin ^{-1} x+c ) D. ( x-sin ^{-1} x+c ) | 12 |
| 1262 | Evaluate ( int e^{x}(tan x-log cos x) d x ) | 12 |
| 1263 | Evaluate ( int frac{sec ^{2} x}{3+tan x} d x ) | 12 |
| 1264 | ( int_{0}^{frac{pi}{2}} frac{sin x-cos x}{1+sin x cdot cos x} d x ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( frac{pi}{4} ) ( c cdot frac{pi}{2} ) D. | 12 |
| 1265 | (1995) then constants A and B are 8. If f(x) = A sin( TX + B, FC)=12 and server = 24, then constants A and B are (a) and (C) O and 4 (d) and o | 12 |
| 1266 | 27. The solution for x of the equation “INOP-1 3 is (2007) at is [2007] (a) v3 (b) 272 ( 2 () None | 12 |
| 1267 | What is ( I_{1} ) equal to? A ( cdot frac{pi}{24} ) B. ( frac{pi}{18} ) c. ( frac{pi}{12} ) D. ( frac{pi}{6} ) | 12 |
| 1268 | ( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x}), boldsymbol{f}(mathbf{0})=mathbf{1}, ) then ( int frac{d x}{f(x)+f(-x)} ) ( mathbf{A} cdot log left(e^{2 x}+1right)+C ) B. ( log left(e^{x}+e^{-x}right)+C ) c. ( tan ^{-1}left(e^{x}right)+C ) D. None | 12 |
| 1269 | Evaluate the integral ( int_{1}^{2} sqrt{(x-1)(2-x)} d x ) A. ( frac{pi}{8} ) в. c. ( frac{1}{8} ) D. | 12 |
| 1270 | Differentiate the following function with respect to ( x ) ( left(2 x^{2}-3right) sin x ) A ( cdot 4 x sin x-left(2 x^{2}-3right) cos x ) B. ( 4 x sin x+left(2 x^{2}+3right) cos x ) c. ( 4 x sin x+left(2 x^{2}-3right) cos x ) D. None of the above | 12 |
| 1271 | The value of ( lim _{n rightarrow infty} frac{(n !) frac{1}{n}}{n} ) is? ( mathbf{A} cdot mathbf{1} ) в. ( frac{1}{e^{2}} ) c. ( frac{1}{2 e} ) D. | 12 |
| 1272 | Find the values of ( c ) that satisfy the MVT for integrals on [-2,3] ( boldsymbol{f}(boldsymbol{t})=mathbf{8} boldsymbol{t}+boldsymbol{e}^{-boldsymbol{3} boldsymbol{t}} ) В. ( c=-0.0973 ) c. ( c=1.0973 ) D. ( c=0.0973 ) | 12 |
| 1273 | Evaluate : ( int x^{3} sqrt{1-x^{8}} d x ) | 12 |
| 1274 | – dx = 2+1 (2006 – 3M, -1) © V2x*-2×2 +1 +(a) v2x4222 +1. | 12 |
| 1275 | If ( frac{mathbf{x}^{2}}{left(mathbf{x}^{2}+mathbf{1}right)left(mathbf{x}^{2}+mathbf{2}right)}=frac{mathbf{A} mathbf{x}+mathbf{B}}{mathbf{x}^{2}+mathbf{1}}+ ) ( frac{mathbf{C x}+mathbf{D}}{mathbf{x}^{2}+mathbf{2}} operatorname{then}(boldsymbol{A}, boldsymbol{C})= ) A. (1,-1) в. (1,1) D. (1,2) | 12 |
| 1276 | Integral of ( frac{left(4 x^{2}-2 sqrt{x}right)}{x}+frac{1}{1+x^{2}}- ) 5 ( operatorname{cosec}^{2} x ) is | 12 |
| 1277 | The integral ( int_{0}^{pi} x f(sin x) d x ) is equal to This question has multiple correct options ( mathbf{A} cdot frac{pi}{2} int_{0}^{pi} f(sin x) d x ) B ( cdot frac{pi}{4} int_{0}^{pi} f(sin x) d x ) ( ^{mathbf{C}} pi int_{0}^{pi / 2} f(sin x) d x ) ( ^{mathrm{D}} pi int_{0}^{pi / 2} f(cos x) d x ) | 12 |
| 1278 | ( intleft(a-a^{n x}right) d x= ) A. ( a x-frac{a^{n x}}{n log a}+c ) в. ( a x+frac{a^{n x}}{n log a}+c ) c. ( a x+frac{a^{n x}}{log a}+c ) D. ( a x+frac{a^{n x+1}}{log a}+c ) | 12 |
| 1279 | Simplify:( intleft(frac{1}{(ln x)}-frac{1}{(ln x)^{2}}right) d x ) | 12 |
| 1280 | Evaluate ( intleft(frac{1}{7}-frac{1}{y^{5 / 4}}right) d y ) A ( cdot_{I}=frac{y}{7}+frac{4}{y^{1 / 4}}+c ) B. ( I=-frac{y}{7}+frac{4}{y^{1 / 4}}+c ) ( ^{mathrm{C}} cdot_{I}=frac{y}{7}-frac{4}{y^{1 / 4}}+c ) D. None of these | 12 |
| 1281 | Let ( f(x) ) be a function satisfying ( f^{prime}(x)=f(x)=e^{x} ) with ( f(0)=1 ) and ( g(x) ) be a function that satisfies ( f(x)+ ) ( g(x)=x^{2} . ) Then, the value of the integral ( int_{0}^{1} f(x) g(x) d x ) is ( ^{A} cdot_{e}+frac{e^{2}}{2}-frac{3}{2} ) B. ( _{e}-frac{e^{2}}{2}-frac{3}{2} ) c. ( _{e}+frac{e^{2}}{2}+frac{5}{2} ) D. ( _{e}-frac{e^{2}}{2}-frac{5}{2} ) | 12 |
| 1282 | Evaluate: ( int_{0}^{3} x^{2}+2 x d x ) | 12 |
| 1283 | ( int frac{boldsymbol{x}^{2}+mathbf{1}}{boldsymbol{x}^{4}+mathbf{1}} boldsymbol{d} boldsymbol{x}= ) ( mathbf{A} cdot frac{1}{sqrt{2}} tan ^{-1}left(frac{x^{2}+1}{sqrt{2 x}}right)+c ) ( mathbf{B} cdot tan ^{-1}left(frac{x^{2}+1}{sqrt{2 x}}right)+c ) ( mathbf{C} cdot frac{1}{sqrt{2}} tan ^{-1}left(frac{x^{2}-1}{sqrt{2 x}}right)+c ) ( mathbf{D} cdot tan ^{-1}left(frac{x^{2}-1}{sqrt{2 x}}right)+c ) | 12 |
| 1284 | Solve:( int frac{2 x+5}{x^{2}+5 x+6} d x ) | 12 |
| 1285 | ( n stackrel{L t}{rightarrow} inftyleft{frac{1}{2 n+1}+frac{1}{2 n+2}+right. ) ( left.frac{1}{2 n+3} cdots+frac{1}{2 n+n}right} ) A. ( log _{e}left(frac{1}{3}right) ) B. ( log _{e}left(frac{2}{3}right) ) c. ( log _{e}left(frac{3}{2}right) ) D ( cdot log _{e}left(frac{4}{3}right) ) | 12 |
| 1286 | ( int e^{x}left(frac{x^{4}+x^{2}+1}{x^{2}+x+1}right) d x= ) A ( cdot e^{x}left(x^{4}+x^{2}+1right)+c ) B ( cdot e^{x}left(x^{2}+x+1right)+c ) c. ( e^{x}left(x^{2}-3 x+4right)+c ) D. ( e^{x}left(x^{2}-4 x+5right)+c ) | 12 |
| 1287 | Integrate the function ( sqrt{x^{2}+4 x+6} ) | 12 |
| 1288 | ( int e^{x}left(frac{2+sin 2 x}{1+cos 2 x}right) d x= ) A ( cdot e^{x} cot x+c ) B ( cdot 2 e^{x} sec ^{2} x+c ) ( mathbf{c} cdot e^{x} cos 2 x+c ) D. ( e^{x} tan x+c ) | 12 |
| 1289 | 5. For any integer n the integral — ſecos-* cos(2n +1)xdx has the value (1985 – 2 Marks) (a) a (c) O (b) 1 (d) none of these | 12 |
| 1290 | Evaluate: ( int_{0}^{pi} frac{x}{a^{2} cos ^{2} x+b^{2} sin ^{2} x} d x ) ( mathbf{A} cdot frac{pi^{2}}{a b} ) B. ( frac{pi^{2}}{2 a b} ) ( mathbf{C} cdot frac{2 pi^{2}}{a b} ) D. ( frac{pi^{2}}{4 a b} ) | 12 |
| 1291 | If ( I=int sec ^{2} x operatorname{cosec}^{4} x d x=A cot ^{3} x+ ) ( B tan x+C cot x+D ) then This question has multiple correct options A ( cdot A=-frac{1}{3} ) в. ( B=2 ) c. ( C=-2 ) D. none of these | 12 |
| 1292 | ( int frac{cos x+sin x}{cos x-sin x} d x ) ( mathbf{A} cdot log sin (pi / 4+x) ) ( mathbf{B} cdot log sec (pi / 4+x) ) ( mathbf{C} cdot log cos (pi / 4+x) ) D ( cdot log sec (pi / 4-x) ) | 12 |
| 1293 | Solve: ( int frac{1}{xleft(x^{4}-1right)} d x ) A. ( -frac{1}{4} ln left|frac{x^{4}-1}{x^{4}}right|+c ) B ( cdot frac{1}{4} ln left|frac{x^{4}-1}{x^{4}}right|+c ) c. ( -frac{1}{4} ln left|frac{x^{2}-1}{x^{2}}right|+c ) D. ( frac{1}{4} ln left|frac{x^{2}-1}{x^{2}}right|+c ) | 12 |
| 1294 | If ( int frac{2 x^{2}+3}{left(x^{2}-1right)left(x^{2}+4right)} d x= ) ( operatorname{Alog}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+boldsymbol{B} tan ^{-1}left(frac{boldsymbol{x}}{mathbf{2}}right)+boldsymbol{C} ) then ( (A, B) ) is A ( cdotleft(-frac{1}{2}, frac{1}{2}right) ) B ( cdotleft(frac{1}{2},-frac{1}{2}right) ) ( left(frac{1}{2}, frac{1}{2}right) ) D. (1,-1) | 12 |
| 1295 | Solve: ( int x^{2} sin ^{2} x d x ) | 12 |
| 1296 | ( int frac{(boldsymbol{x}+mathbf{1})}{boldsymbol{x}left(mathbf{1}+boldsymbol{x} boldsymbol{e}^{boldsymbol{x}}right)} boldsymbol{d} boldsymbol{x}= ) ( mathbf{A} cdot log left|frac{e^{x}}{1+x e^{x}}right|+c ) в. ( -log left|frac{e^{x}}{1+x e^{x}}right|+c ) c. ( log left|frac{x e^{x}}{1+x e^{x}}right|+c ) D. ( -log left|frac{e^{x}}{1-x e^{x}}right|+c ) | 12 |
| 1297 | ( int frac{cot sqrt{x}}{2 sqrt{x}} d x ) is equal to ( =-ldots+C ) B . ( log |sin sqrt{x}| ) c. ( frac{1}{2} log |sin sqrt{x}| ) D. None of these | 12 |
| 1298 | ( intleft(frac{e^{5 log x}-e^{4 log x}}{e^{3 log x}-e^{2 log x}}right) d x= ) A . ( x+c ) B. ( 3 x^{2}+c ) c. ( frac{x^{3}}{3}+c ) D. ( frac{x^{3}}{2}+c ) | 12 |
| 1299 | ( int x^{2} tan ^{-1} x d x ) A. ( frac{x^{3}}{3} tan ^{-1} x-frac{1}{6} x^{2}+frac{1}{6} log left(x^{2}+1right) ) B. ( frac{x^{3}}{3} tan ^{-1} x+frac{1}{6} x^{2}+frac{1}{6} log left(x^{2}+1right) ) C ( frac{x^{3}}{3} tan ^{-1} x-frac{1}{3} x^{2}+frac{1}{6} log left(x^{2}+1right) ) D. ( frac{x^{3}}{3} tan ^{-1} x-frac{1}{6} x^{2}+frac{1}{3} log left(x^{2}+1right) ) | 12 |
| 1300 | Number of positive continuous functions ( f(x) ) defined in [0,1] for which ( int_{0}^{1} f(x) d x=1, int_{0}^{1} x f(x) d x=2 ) ( int_{0}^{1} x^{2} f(x) d x=4 ) ( A ) B. 4 c. Infinite D. None of these | 12 |
| 1301 | ( int frac{x^{4}}{x^{2}+1} d x ) | 12 |
| 1302 | Evaluate: ( int e^{x^{3}+x^{2}-1}left(3 x^{4}+2 x^{3}+2 xright) d x ) | 12 |
| 1303 | Evaluate ( int frac{x+1}{x^{2}+3 x+12} d x ) | 12 |
| 1304 | Show that: ( int_{0}^{frac{pi}{4}} log (1+tan x) d x= ) ( frac{pi}{8} log 2 ) | 12 |
| 1305 | Evaluate ( int sin ^{-1} frac{2 x}{1+x^{2}} d x ) | 12 |
| 1306 | Solve : ( int tan ^{2}(2 x-3) d x ) | 12 |
| 1307 | Integrate the following functions with respect to t: ( int frac{boldsymbol{d t}}{(boldsymbol{6 t}-mathbf{1})} ) A ( cdot frac{1}{6} ln (6 t-1)+C ) B ( cdot ln (6 t-1)+C ) c. ( -ln (6 t-1)+C ) D. ( -frac{1}{6} ln (6 t-1)+C ) | 12 |
| 1308 | ( int(a x+b)^{2} d x ) | 12 |
| 1309 | ( int frac{x^{4}}{(x+2)left(x^{2}+1right)} d x ) How to change this improper function to Rational function. | 12 |
| 1310 | ( frac{7 x^{3}+3 x^{2}-x+1}{x+1}=left(a x^{2}+b x+cright) ) ( frac{2}{x+1} ) then ( a= ) A . 3 B. 7 ( c cdot 1 ) D. | 12 |
| 1311 | Find: ( int frac{x e^{x}}{(1+x)^{2}} d x ) | 12 |
| 1312 | ( int_{2}^{3}(1+2 x) d x ) | 12 |
| 1313 | ( int x^{5} d x ) | 12 |
| 1314 | Solve: ( int cos ^{3} x d x ) | 12 |
| 1315 | If ( I=int frac{d x}{sqrt{(1-x)(x-2)}}, ) then ( I ) is equal to ( A cdot sin ^{-1}(2 x-3)+C ) B. ( sin ^{-1}(2 x+5)+C ) ( c cdot sin ^{-1}(3-2 x)+C ) D ( cdot sin ^{-1}(5-2 x)+C ) | 12 |
| 1316 | ( mathbf{f} boldsymbol{I}=int cot ^{-1}left(frac{boldsymbol{a}^{2}-boldsymbol{a} boldsymbol{x}+boldsymbol{x}^{2}}{boldsymbol{a}^{2}}right) boldsymbol{d} boldsymbol{x}, ) then equals ( mathbf{A} cdot_{x tan ^{-1}}left(frac{x}{a}right)-(x-a) tan ^{-1}left(frac{x-a}{a}right)+C ) B ( cdot frac{a}{2} log left(2 a^{2}-2 a x+x^{2}right)-frac{a}{2} log left(x^{2}+a^{2}right)+C ) C ( cdot x tan ^{-1}left(frac{x}{a}right)+(x-a) tan ^{-1}left(frac{x-a}{a}right)+frac{a}{2} log left(2 a^{2}-right. ) ( left.2 a x+x^{2}right)+C ) D. none of these | 12 |
| 1317 | Resolve ( frac{x^{4}}{(x-1)^{4}(x+1)} ) into partia fractions. A ( cdot frac{1}{2(x-1)^{4}}-frac{7}{4(x-1)^{3}}+frac{17}{8(x-1)^{2}}+frac{15}{16(x-1)}+frac{1}{16} ) ( frac{1}{(x+1)} ) B. ( frac{1}{2(x-1)^{4}}+frac{7}{4(x-1)^{3}}+frac{17}{8(x-1)^{2}}+frac{15}{16(x-1)}+frac{1}{16} ) ( frac{1}{(x+1)} ) C ( frac{1}{2(x-1)^{4}}+frac{5}{4(x-1)^{3}}+frac{17}{8(x-1)^{2}}+frac{15}{16(x-1)}+frac{1}{16} ) ( frac{1}{(x+1)} ) D ( frac{1}{2(x-1)^{4}}+frac{7}{4(x-1)^{3}}+frac{13}{8(x-1)^{2}}+frac{15}{16(x-1)}+frac{1}{16} ) ( frac{1}{(x+1)} ) | 12 |
| 1318 | +…+. Show that : lim – +- n+on+1 n +2 -) = log 6 (1091 2M | 12 |
| 1319 | Evaluate: ( int sec ^{4} x cdot operatorname{cosec}^{2} x d x ) A. ( frac{1}{3} t^{3}+t ) B ( cdot frac{1}{3} t^{3}+2 t-frac{1}{t} ) c. ( frac{1}{2} t^{3}+2 t-frac{1}{t} ) D. ( frac{1}{3} t^{3}-t-frac{1}{t} ) | 12 |
| 1320 | ( int_{a / 4}^{3 a / 4} frac{sqrt{x}}{sqrt{a-x}+sqrt{x}} d x ) is equal to? A ( cdot frac{a}{4} ) в. ( c .-a ) D. none of these | 12 |
| 1321 | Integrate the function ( x log x ) | 12 |
| 1322 | ( frac{x^{2}}{left(x^{2}+a^{2}right)left(x^{2}+b^{2}right)}= ) ( kleft[frac{a^{2}}{x^{2}+a^{2}}-frac{b^{2}}{x^{2}+b^{2}}right] Rightarrow k= ) ( A ) в. ( frac{1}{a^{2}+b^{2}} ) c. ( frac{1}{a^{2}-b^{2}} ) D. ( frac{1}{b^{2}-a^{2}} ) | 12 |
| 1323 | ( int_{0}^{1} tan ^{-1}left[frac{2 x-1}{1+x-x^{2}}right] d x= ) A . B. ( 1 / 2 ) ( c ) D . ( pi / 6 ) | 12 |
| 1324 | Evaluate ( int r^{4}left(7-frac{r^{5}}{10}right) d r ) | 12 |
| 1325 | Match the integrals of ( f(x) ) if | 12 |
| 1326 | Evaluate the given integral. ( int x^{2} cos x d x ) | 12 |
| 1327 | Integrate ( int x cos ^{-1} x d x ) | 12 |
| 1328 | ( int_{0}^{2} 3 x+2 d x ) | 12 |
| 1329 | ( int_{0}^{pi / 4} frac{tan ^{3} x}{1+cos 2 x} d x ) | 12 |
| 1330 | Evaluate: ( int frac{x e^{x}}{(x+1)^{2}} d x ) | 12 |
| 1331 | Evaluate the integral ( int_{0}^{2 pi} frac{1}{1+tan ^{4} x} d x ) A. в. c. ( frac{3 pi}{4} ) D. ( pi ) | 12 |
| 1332 | ( int_{0}^{pi / 2} frac{d x}{a^{2} cos ^{2} x+b^{2} sin ^{2} x} ) equals- A . ( pi / a b ) B. ( 2 pi / a b ) c. ( a b / pi ) D. ( pi / 2 a b ) | 12 |
| 1333 | ( int frac{3.2^{x}-2.3^{x}}{2^{x}} d x= ) A ( cdot 3 x+frac{2(1.5)^{x}}{log (1.5)}+c ) B. ( 3 x-frac{2(1.5)^{x}}{log (1.5)}+c ) ( mathbf{c} cdot 3 x-2(1.5)^{x} log 1.5+c ) D. ( 3 x+2 log 1.5(1.5)^{x}+c ) | 12 |
| 1334 | Evaluate the following definite integral: ( int_{-pi / 4}^{pi / 4} log (cos x+sin x) d x ) A . ( pi log 2 ) B. – pi log2 C ( cdot-frac{pi}{4} log 2 ) D. ( pi^{2} log 2 ) | 12 |
| 1335 | 5. sin xdx The value of 21 – @) x+log cos – 5) i+c (b) x-log|sin(x-4) 1+c (©) x+log|sin(x-4) 1+c (W) x-log/cos(x-4) i to S | 12 |
| 1336 | Integrate the function ( sqrt{1-4 x-x^{2}} ) | 12 |
| 1337 | Find the values of ( c ) that satisfy the MVT for integrals on ( [mathbf{0}, mathbf{1}] ) ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(mathbf{1}-boldsymbol{x}) ) A ( c_{c}=frac{1}{2} ) B. ( c=-frac{1}{2} ) c. ( _{c=frac{2}{3}} ) D. ( c=-frac{1}{3} ) | 12 |
| 1338 | Find the following integral. ( int e^{x}left(sec ^{2} x+tan xright) cdot d x ) | 12 |
| 1339 | ( int frac{e^{x}}{x}left(x cdot(log x)^{2}+2 log xright) d x ) | 12 |
| 1340 | If ( b>a, ) and ( I=int_{a}^{b} sqrt{frac{x-a}{b-x}} d x, ) then ( I ) equals A ( cdot frac{pi}{2}(b-a) ) в. ( pi(b-a) ) c. ( pi / 2 ) D. ( 2 pi(b-a) ) | 12 |
| 1341 | Evaluate the given integral. ( int frac{5 x+3}{sqrt{x^{2}+4 x+10}} d x ) | 12 |
| 1342 | ( int_{0}^{infty} boldsymbol{f}left(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}right) cdot frac{ln boldsymbol{x}}{boldsymbol{x}} boldsymbol{d} boldsymbol{x} ) A. Is equal to zero B. Is equal to one ( ^{mathrm{c}} ). ( _{text {Is equal to }} frac{1}{2} ) D. can not be evaluated | 12 |
| 1343 | Solve: ( int frac{1}{x} sqrt{frac{x-1}{x+1}} d x ) | 12 |
| 1344 | Solve : ( int frac{x}{2 x-3} d x ) | 12 |
| 1345 | Evaluate ( int frac{1+x^{-2 / 3}}{1+x} d x ) The ans is ( =frac{1}{2}left[log (t+1)^{4}left(t^{2}-t+right.right. ) 1) ( ]+sqrt{3} cdot tan ^{-1} frac{2 t-1}{sqrt{(3))}} ) then ( t=? ) ( mathbf{A} cdot x^{1 / 3} ) B . ( x^{1 / 2} ) ( c ) ( mathbf{D} cdot x^{3} ) | 12 |
| 1346 | ( boldsymbol{I}=int frac{1}{boldsymbol{x}(1+log boldsymbol{x})} cdot boldsymbol{d} boldsymbol{x} ) | 12 |
| 1347 | Solve ( int frac{x^{2}-1}{(x+1)} d x ) | 12 |
| 1348 | ( mathbf{f} boldsymbol{alpha}^{2}+boldsymbol{beta}^{2}+boldsymbol{gamma}^{2}=mathbf{1}, ) then highest integral value of ( boldsymbol{alpha} boldsymbol{beta}+boldsymbol{beta} boldsymbol{gamma}+boldsymbol{alpha} boldsymbol{gamma} ) is | 12 |
| 1349 | ( int frac{4 e^{x}+6 e^{-x}}{9 e^{x}-4 e^{-x}} d x=A x+B log _{e}left(9 e^{2 x}-4right) ) ( A=_{—-}, B=_{—-}, C= ) | 12 |
| 1350 | Evaluate: ( int_{0}^{frac{pi}{4}}[sqrt{tan x}+sqrt{cot x}] d x ) A ( cdot frac{pi}{sqrt{2}} ) в. c. ( frac{3 pi}{sqrt{2}} ) D. | 12 |
| 1351 | Evaluate: ( int_{a}^{b} x d x ) using limit of sum. A ( cdot frac{b^{2}-a^{2}}{3} ) B. ( frac{b^{2}+a^{2}}{2} ) c. ( frac{b^{2}-a^{2}}{2} ) D. None of these | 12 |
| 1352 | Solve ( int frac{1}{x^{2} cos ^{2}(1 / x)} d x ) A . ( tan x ) B. – tan ( x ) ( c cdot cot x ) D. – co | 12 |
| 1353 | ( int_{0}^{1} frac{1}{1+x} d x= ) ( A cdot log 2 ) B. ( frac{1}{2} log 2 ) ( c cdot 2 ) D. ( log 3 ) | 12 |
| 1354 | The acceleration of a particle varies with time ( t ) seconds according to the relation ( a=6 t+6 m s^{-2} . ) Find velocity and position as functions of time. It is given that the particle starts from origin at ( t=0 ) with velocity ( 2 m s^{-1} ) | 12 |
| 1355 | ſa-x50,100 dx 49. The value of 50500 [(1 – x 50,101dx – is. | 12 |
| 1356 | ( int frac{x^{2}-8 x+7}{left(x^{2}-3 x-10right)^{2}} d x= ) ( boldsymbol{P} log |boldsymbol{x}-mathbf{5}|+boldsymbol{Q} frac{mathbf{1}}{boldsymbol{x}-mathbf{5}}- ) ( boldsymbol{R} cdot log |boldsymbol{x}+mathbf{2}|-boldsymbol{S} cdot frac{mathbf{1}}{boldsymbol{x}+mathbf{2}}+boldsymbol{c} . ) Then A ( cdot P=-frac{45}{98} ) в. ( Q=frac{8}{49} ) ( c cdot R=frac{15}{49} ) D. All of these | 12 |
| 1357 | If ( int e^{x}(1+x) sec ^{2}left(x e^{x}right) d x=f(x)+c ) then ( boldsymbol{f}(boldsymbol{x})= ) A ( cdot cos left(x e^{x}right) ) B. ( sin left(x e^{x}right) ) c. ( 2 tan ^{-1}(x) ) D. ( tan left(x e^{x}right) ) | 12 |
| 1358 | Solve: ( int frac{sqrt{x}-sqrt{a}}{sqrt{x+a}} d x ) | 12 |
| 1359 | Find the value of ( int_{-pi}^{pi} frac{cos ^{2} x}{1+a^{x}}, a>0 ) A . ( 1 pi ) в. ( pi / 2 ) c. ( pi / 4 ) D. ( 2 pi ) | 12 |
| 1360 | ( mathrm{f} frac{d y}{d x}+sqrt{frac{1-y^{2}}{1-x^{2}}}=0 . ) Prove that, ( boldsymbol{x} sqrt{1-boldsymbol{y}^{2}}+boldsymbol{y} sqrt{1-boldsymbol{x}^{2}}=boldsymbol{A} ) where ( mathbf{A} ) is constant. | 12 |
| 1361 | Assertion STATEMENT-1: If ( f(x) ) is continuous on ( [a, b], ) then there exists a point ( c in(a, b) ) such that ( int_{a}^{b} f(x) d x=f(c)(b-a) ) Reason STATEMENT-2: For ( a<b ), if ( m ) and ( M ) are, respectively, the smallest and greatest values of ( boldsymbol{f}(boldsymbol{x}) ) on ( [boldsymbol{a}, boldsymbol{b}] ) ( operatorname{then} m(b-a) leq int_{a}^{b} f(x) d x leq(b- ) ( boldsymbol{a}) boldsymbol{M} ) A. STATEMENT-1 is True, STATEMENT-2 is True STATEMENT-2 is a correct explanation for STATEMENT- 1 B. STATEMENT-1 is True, STATEMENT-2 is True STATEMENT-2 is NOT a correct explanation for STATEMENT-1. C . STATEMENT-1 is True, STATEMENT-2 is False D. STATEMENT-1 is False, STATEMENT-2 is True | 12 |
| 1362 | ( int_{0}^{1 / 2} e^{x}left[sin ^{-1} x+frac{1}{sqrt{1-x^{2}}}right] d x= ) ( ^{A} cdot frac{e^{4}}{4} ) B. ( frac{pi sqrt{6}}{6} ) ( c cdot frac{sqrt{pi}}{_{1}} ) D. ( frac{pi sqrt{6}}{2} ) | 12 |
| 1363 | ( int frac{1}{sqrt{1+x}} d x ) | 12 |
| 1364 | 50. The value of || cos xß dx is: [JEE M 2019-9 Jan (M) | 12 |
| 1365 | ( int frac{e^{x}}{e^{2 x}+5 e^{x}+6} d x= ) A ( cdot log left|frac{e^{x}+2}{e^{x}+3}right|+c ) в. ( log left|frac{e^{x}+3}{e^{x}+2}right|+c ) c. ( log left|frac{e^{x}-2}{e^{x}-3}right|+c ) D. ( log left|frac{e^{x}-3}{e^{x}-2}right|+c ) | 12 |
| 1366 | pr 2x(1+sin x) 33. Determine the value of |”. J-TT 1+cOS X. | 12 |
| 1367 | Solve ( int frac{3 x-1}{(x+2)^{2}} d x ) | 12 |
| 1368 | The value of ( 2 int sin x operatorname{cosec} 4 x d x ) is equal to: ( ^{mathbf{A}} cdot frac{1}{2 sqrt{2}} ln left|frac{1+sqrt{2} sin x}{1-sqrt{2} sin x}right|-frac{1}{4} ln left|frac{1+sin x}{1-sin x}right|+c ) ( frac{1}{2 sqrt{2}} ln left|frac{1+sqrt{2} sin x}{1-sqrt{2} sin x}right|-frac{1}{2} ln left|frac{1+sin x}{cos x}right|+c ) ( frac{1}{2 sqrt{2}} ln left|frac{1-sqrt{2} sin x}{1+sqrt{2} sin x}right|-frac{1}{4} ln left|frac{1+sin x}{1-sin x}right|+c ) ( frac{1}{2 sqrt{2}} ln left|frac{1-sqrt{2} sin x}{1+sqrt{2} sin x}right|+frac{1}{2} ln left|frac{1+sin x}{cos x}right|+c ) | 12 |
| 1369 | ( int frac{3 x+1}{(x-1)^{2}(x+3)} d x= ) A. ( log left|frac{x-1}{x+3}right|-frac{1}{x-1}+c ) в. ( frac{1}{2} log left|frac{x-1}{x+3}right|-frac{1}{x-1}+c ) c. ( frac{1}{2} log left|frac{x-1}{x+3}right|+frac{1}{x+1}+c ) D. ( frac{1}{2} log left|frac{x+1}{x-3}right|+frac{1}{x+1}+c ) | 12 |
| 1370 | Solve ( boldsymbol{I}=int frac{1}{cos ^{2} x(1-tan x)^{2}} d x ) A ( cdot_{I}=frac{-1}{1-cot x}+C ) B. ( I=frac{1}{1-tan x}+C ) ( ^{c} I=frac{-1}{1-tan x}+C ) D. None of these | 12 |
| 1371 | ( int_{1}^{4} frac{mathbf{x} mathbf{d x}}{sqrt{mathbf{2 + 4 x}}}= ) A ( cdot frac{1}{2} ) B. ( frac{1}{sqrt{2}} ) ( c cdot frac{3}{2} ) D. ( frac{3}{sqrt{2}} ) | 12 |
| 1372 | Integrate ( int_{0}^{1}left(3 x^{2}+2 xright) d x ) | 12 |
| 1373 | Integrate the function ( frac{1}{sqrt{9-25 x^{2}}} ) | 12 |
| 1374 | ( int frac{x}{sqrt{9+8 x-x^{2}}} d x ) is equal to A ( cdot-sqrt{9+8 x-x^{2}}+4 sin ^{-1}left(frac{x-4}{5}right)+C ) B. ( -sqrt{9+8 x-x^{2}}+4 cos ^{-1}left(frac{x-4}{5}right)+C ) c. ( -sqrt{9+8 x-x^{2}}+4 cos ^{-1}left(frac{x-3}{2}right)+C ) D ( cdot sqrt{9+8 x-x^{2}}+4 sin ^{-1}left(frac{x-4}{5}right)+C ) | 12 |
| 1375 | Evaluate the integral ( int_{0}^{1} frac{sin ^{-1} x}{x} d x ) ( A cdot pi log 2 ) в. ( -pi log 2 ) C ( cdot-frac{pi}{2} log 2 ) D. ( frac{pi}{2} log 2 ) | 12 |
| 1376 | The value of ( int x^{3} log x d x ) is A ( cdot frac{1}{16}left(4 x^{4} log x-x^{4}+cright) ) B ( cdot frac{1}{8}left(x^{4} log x-4 x^{4}+cright) ) c. ( frac{1}{16}left(4 x^{4} log x+x^{4}+cright) ) ( frac{x^{4} log x}{4}+c ) | 12 |
| 1377 | Solve : ( int frac{x^{2}+3 x-1}{(x+1)^{2}} d x ) | 12 |
| 1378 | 1/2 14. If | xf (sin x)dx = A ( f (sin x)dx, then A is 12004 (a) 20 (6) (d) 0 | 12 |
| 1379 | Evaluate ( : int frac{1}{sqrt{(x-1)(x-2)}} d x ) A. [ begin{array}{l}text { B. } log left|left(x-frac{3}{2}right)+sqrt{x^{2}-3 x+2}right|+C \ text { c. } log left(left(x-frac{3}{2}right)+sqrt{x^{2}-3 x+2}right)+C \ text { D. } & =log left|left(x-frac{3}{2}right)+sqrt{x^{2}+3 x+2}right|+Cend{array} ] | 12 |
| 1380 | The value ( sqrt{2} int frac{sin x d x}{sin left(x-frac{pi}{4}right)} ) is A ( cdot x-log left|sin left(x-frac{pi}{4}right)right|+C ) B. ( x+log left|sin left(x-frac{pi}{4}right)right|+C ) c. ( x-log left|cos left(x-frac{pi}{4}right)right|+C ) D. ( x+log left|cos left(x-frac{pi}{4}right)right|+C ) | 12 |
| 1381 | Integrate ( int_{0}^{2}left(x^{2}+xright) d x ) | 12 |
| 1382 | If ( phi(x)=phi^{prime}(x) ) and ( phi(1)=2 ) then ( phi(3) ) is equal to ( A cdot phi^{2} ) B . ( 2 phi^{2} ) ( c cdot 3 phi^{2} ) D. ( 2 phi^{3} ) | 12 |
| 1383 | If an antiderivative of ( f(x) ) is ( e^{x} ) and that of ( g(x) ) is ( cos x, ) then ( int f(x) cos x d x+int g(x) e^{x} d x= ) ( mathbf{A} cdot f(x) g(x)+c ) в. ( f(x)+g(x)+c ) c. ( e^{x} cos x+c ) D. ( -e^{x} cos x+c ) | 12 |
| 1384 | By using the properties of definite integrals, evaluate the integral ( int_{0}^{frac{pi}{2}} cos ^{2} x d x ) | 12 |
| 1385 | Solve ( left[-int_{0}^{pi / 2} cos left(frac{pi}{4}+frac{x}{2}right) e^{x}right] d x ) | 12 |
| 1386 | Evaluate the following definite integral: ( int_{pi / 6}^{pi / 4} operatorname{cosec} x d x ) | 12 |
| 1387 | Evaluate ( int_{0}^{5} x^{4} d x ) | 12 |
| 1388 | Solve ( : int_{0}^{1} x^{2}left(1-x^{2}right)^{3 / 2} d x= ) | 12 |
| 1389 | ( int cos x sqrt{4-sin ^{2} x} d x ) A. ( frac{t}{2} sqrt{4-t^{2}}-frac{1}{2} cdot 4 sin ^{-1} frac{t}{2} ) B ( cdot sqrt{4-t^{2}}+4 sin ^{-1} frac{t}{2} ) C ( cdot frac{t}{2} sqrt{4-t^{2}}+frac{1}{2} cdot 4 sin ^{-1} frac{t}{2} ) D ( cdot frac{t}{2} sqrt{4+t^{2}}+frac{1}{2} cdot 4 sin ^{-1} frac{t}{2} ) | 12 |
| 1390 | ( int_{0}^{pi / 2} cos ^{2} x d x ) | 12 |
| 1391 | ( int frac{(x-1) e^{x}}{(x+1)^{3}} d x ) is A ( cdot frac{e^{x}}{x+1} ) В. ( e^{x}left(frac{x}{x+1}right) ) c. ( frac{e^{x}(x-1)}{(x+1)^{2}} ) D. ( frac{e^{x}}{(x+1)^{2}} ) | 12 |
| 1392 | Evaluate the following integral: ( int sec ^{4} x tan x d x ) | 12 |
| 1393 | 6. continous functions. Then Letf:R → R and g:R → R be continous fun the value of the integral TI/2 J [f(x) + f(-x)][g(x)-g(-x)]dx is 1990 – 0/2 (a) (b) 1 (0) 1 (d) 0 18(x)-8(-x)]dx is 1990 – 2 Marks) | 12 |
| 1394 | Suppose ( M=int_{0}^{pi / 2} frac{cos x}{x+2} d x, N= ) ( int_{0}^{pi / 4} frac{sin x cos x}{(x+1)^{2}} d x . ) Then, the value of ( (M-N) ) equals A ( frac{3}{pi+2} ) в. ( frac{2}{pi-4} ) c. ( frac{4}{pi-2} ) D. ( frac{2}{pi+4} ) | 12 |
| 1395 | Solve : ( boldsymbol{I}=int frac{boldsymbol{x}+mathbf{9}}{boldsymbol{x}^{2}+mathbf{5}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1396 | Solve ( int_{0}^{pi} frac{1}{3+2 sin x+cos x} d x ) A ( cdot frac{5 pi}{4} ) в. ( c cdot-frac{pi}{4} ) D. None of these | 12 |
| 1397 | The value of ( int_{1}^{3} x^{2} d x ) is: A ( cdot frac{26}{3} ) в. ( frac{28}{3} ) c. ( frac{25}{3} ) D. None of these | 12 |
| 1398 | Evaluate : ( int frac{sec ^{2} x}{tan x} d x ) | 12 |
| 1399 | 1U UU UUIIULUID 11. Let I = tan” x dx, (n > 1). 14 +16=a tanº C is constant of integration, then the ordered equal to : >1).14 +16 = a tan x + bx5 + C, where on, then the ordered pair (a, b) is [JEEM 2017] (c) (5,0) | 12 |
| 1400 | If a continuous function ( boldsymbol{f} ) satisfies ( int_{0}^{x^{2}} f(t) d t=x^{2}(1+x) operatorname{then} f(4) ) is equal to ( A cdot 7 ) B. 4 c. 5 ( D ) | 12 |
| 1401 | ntegrate the function ( frac{mathbf{5 x}}{(x+1)left(x^{2}+9right)} ) | 12 |
| 1402 | The value of ( int sqrt{1+sec x} d x ) is ( A cdot sin ^{-1}(sqrt{2} sin x)+C ) B ( cdot 2 sin ^{-1}left(sqrt{2} sin frac{x}{2}right)+C ) c. ( 2 sin ^{-1}(sqrt{2} sin x)+C ) D ( cdot sin ^{-1}left(sqrt{2} sin frac{x}{2}right)+C ) | 12 |
| 1403 | If ( f(x)=int frac{1}{x-sqrt{x^{2}+1}} ) and ( f(0)= ) ( frac{1+sqrt{2}}{2}, ) then ( f(1) ) is equal to ( ^{A} cdot log (sqrt{sqrt{2}-1}) ) B. ( frac{-1}{sqrt{2}} ) c. ( 1+sqrt{2} ) D. ( frac{1}{2} log (1+sqrt{2}) ) | 12 |
| 1404 | Evaluate: ( int tan ^{-1} sqrt{x} d x ) | 12 |
| 1405 | If ( boldsymbol{A}=int_{0}^{pi} frac{cos boldsymbol{x}}{(boldsymbol{x}+mathbf{2})^{2}} boldsymbol{d} boldsymbol{x}, ) then ( int_{0}^{frac{pi}{2}} frac{sin 2 x}{(x+1)} d x ) is equal to A ( cdot frac{1}{2}+frac{1}{pi+2}-A ) B. ( frac{1}{pi+2}-A ) ( c cdot 1+frac{1}{pi+2}-A ) D. ( _{A-} frac{1}{2}-frac{1}{pi+2} ) | 12 |
| 1406 | Evaluate ( : int_{10}^{2}left(x^{2}+x+2right) d x ) | 12 |
| 1407 | ( int frac{d x}{sqrt{2 e^{x}-1}} ) equals to A ( cdot sec ^{-1} sqrt{2 e^{x}}+c ) B・sec-1 ( left(sqrt{2} e^{x}right)+c ) ( mathbf{c} cdot 2 sec ^{-1}left(sqrt{2} e^{x}right)+c ) D. ( 2 sec ^{-1} sqrt{2 e^{x}}+c ) | 12 |
| 1408 | Solve ( int x log x d x ) | 12 |
| 1409 | ( int x^{x} log (e x) d x ) is equal to A ( cdot x^{x}+c ) B. ( x cdot log x+c ) ( mathbf{c} cdot(log x)^{x}+c ) D. ( x^{log x}+c ) | 12 |
| 1410 | Integrate: ( left(frac{a}{sqrt{x}}+2 b sqrt[3]{x^{2}}right) ) w.r.t | 12 |
| 1411 | Find the integral of ( intleft(sqrt{boldsymbol{x}}-frac{mathbf{1}}{sqrt{boldsymbol{x}}}right)^{2} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1412 | Evaluate: ( int x+5 d x ) | 12 |
| 1413 | ( int x^{3} tan ^{-1} x d x ) | 12 |
| 1414 | Integrate ( sec ^{3} x ) w.r.t. ( x ) | 12 |
| 1415 | 14. The integral ſ sec2/3 x cosec 4/3x dx is equal to: JEE M 2019-9 April (M) (a) –3 tan-1/3x+C (b) – tan 43 x +C (C) –3 cot-1/3 x + C (d) 3 tan-1/3 x +C (Here C is a constant of integration) | 12 |
| 1416 | ( int x sin x sec ^{3} x d x= ) A. ( frac{1}{2}left[sec ^{2} x-tan xright]+c ) B. ( frac{1}{2}left[x sec ^{2} x-tan xright]+c ) c. ( frac{1}{2}left[x sec ^{2} x+tan xright]+c ) D. ( frac{1}{2}left[sec ^{2} x+tan xright]+c ) | 12 |
| 1417 | Evaluate ( int frac{x^{3}}{sqrt{1+2 x^{4}}} d x ) | 12 |
| 1418 | If ( f(x) ) is a polynomial satisfying ( boldsymbol{f}(boldsymbol{x}) boldsymbol{f}left(frac{1}{x}right)=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}left(frac{1}{x}right), operatorname{and} boldsymbol{f}(boldsymbol{3})=mathbf{8} mathbf{2} ) then ( int frac{f(x)}{x^{2}+1} d x= ) A ( cdot x^{3}-x+2 tan ^{-1} x+c ) B. ( frac{1}{3} x^{3}-x+tan ^{-1} x+c ) c. ( frac{x^{3}}{3}-x+2 tan ^{-1} x+c ) D. ( frac{1}{3} x^{3}+x+2 tan ^{-1} x+c ) | 12 |
| 1419 | Evaluate the following: ( int(1+x) e^{x} d x ) | 12 |
| 1420 | If ( frac{mathbf{3} boldsymbol{x}+mathbf{4}}{boldsymbol{x}^{2}-mathbf{3} boldsymbol{x}+mathbf{2}}=frac{boldsymbol{A}}{boldsymbol{x}-mathbf{2}}-frac{boldsymbol{B}}{boldsymbol{x}-mathbf{1}}, ) then ( (A, B)= ) A. (7,10) () B. (10,7) c. (10,-7) D. (-10,7) | 12 |
| 1421 | The number of integral solutions ( (x, y) ) of the equations ( boldsymbol{x} sqrt{boldsymbol{y}}+boldsymbol{y} sqrt{boldsymbol{x}}=mathbf{2 0} ) and ( boldsymbol{x} sqrt{boldsymbol{x}}+boldsymbol{y} sqrt{boldsymbol{y}}=boldsymbol{6 5} ) is: ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. None of these | 12 |
| 1422 | Solve: ( int frac{x+sqrt{1-x^{2}}}{x sqrt{1-x^{2}}} d x=? ) | 12 |
| 1423 | Evaluate ( : int frac{cos ^{-1} x}{x^{2}} d x ) | 12 |
| 1424 | ( int frac{sqrt{x^{2}+1}left[log left(x^{2}+1right)-2 log xright]}{x^{4}} ) is equal to A. [ frac{1}{3}left(1+frac{1}{x^{2}}right)^{frac{3}{2}}left[log left(1+frac{1}{x^{2}}right)+frac{2}{3}right]+C ] в. ( quad-frac{1}{3}left(1+frac{1}{x^{2}}right)^{frac{3}{2}}left[log left(1+frac{1}{x^{2}}right)-frac{2}{3}right]+C ) c. ( quad frac{1}{3}left(1+frac{1}{x^{2}}right)^{frac{1}{2}}left[log left(1+frac{1}{x^{2}}right)+frac{2}{3}right]+C ) D. None of these | 12 |
| 1425 | Evaluate ( int frac{left(1+2 x^{2}right) d x}{x^{2}left(1+x^{2}right)} ) | 12 |
| 1426 | Solve: ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{boldsymbol{x}+mathbf{1}}+sqrt{boldsymbol{x}}} ) | 12 |
| 1427 | Evaluate ( int frac{x^{3}+1}{x^{2}+1} d x= ) | 12 |
| 1428 | Integrate ( int e^{log (sec x+tan x)} sqrt{1+tan ^{2} x} d x ) | 12 |
| 1429 | Solve : ( int frac{d x}{sqrt{x-x^{2}}} ) ( mathbf{A} cdot 2 sin ^{-1} sqrt{x}+c ) B. ( 2 sin ^{-1} x+c ) ( mathbf{c} cdot 2 x sin ^{-1} x+c ) ( mathbf{D} cdot sin ^{-1} sqrt{x}+c ) | 12 |
| 1430 | Solve ( int frac{x e^{x}}{(x+1)^{2}} d x ) A ( cdot frac{e^{x}}{x+1}+C ) в. ( frac{x}{(x+1)^{2}}+C ) c. ( e^{x}(x+1)+C ) D. ( x(x+1)^{2}+C ) | 12 |
| 1431 | If ( omega, omega^{2} ) be the complex cube roots of unity, and ( boldsymbol{f}(boldsymbol{x})= ) [ left|begin{array}{ccc} boldsymbol{x}+mathbf{1} & boldsymbol{omega} & boldsymbol{omega}^{2} \ boldsymbol{omega} & boldsymbol{x}+boldsymbol{omega}^{2} & mathbf{1} \ boldsymbol{omega}^{2} & boldsymbol{1} & boldsymbol{x}+boldsymbol{omega} end{array}right| ] then ( int_{frac{-pi}{2}}^{frac{pi}{2}} f(x) d x ) is equal to? A ( cdot frac{x^{4}}{4} ) в. ( frac{3}{4} x ) c. 0 D. None of the above | 12 |
| 1432 | Integrate the function ( tan ^{-1} sqrt{frac{1-x}{1+x}} ) | 12 |
| 1433 | sec? f(t)dt – equals 31. lim -2 (2007 – 3 marks) se o re 3,7 ca are | 12 |
| 1434 | Evaluate ( int_{1}^{2}left(x^{2}-1right) d x ) | 12 |
| 1435 | The value of ( int frac{sin x+cos x}{sqrt{1+sin 2 x}} d x ) ( mathbf{A} cdot sin x+c ) B. ( x+c ) c. ( cos x+c ) D ( cdot frac{1}{2}(sin x+cos x) ) | 12 |
| 1436 | Solve the equation:( int_{0}^{2}left[x^{2}-x+1right] d x ) | 12 |
| 1437 | If ( boldsymbol{f}(boldsymbol{x})=int_{0}^{sin boldsymbol{x}} cos ^{-1} boldsymbol{t} boldsymbol{d} boldsymbol{t}+ ) ( int_{0}^{cos x} sin ^{-1} t d t, 0<x<frac{pi}{2} ) then ( fleft(frac{pi}{4}right) ) is? A ( cdot frac{pi}{sqrt{2}} ) B ( cdot 1+frac{pi}{2 sqrt{2}} ) c. D. none of these | 12 |
| 1438 | Solve ( int frac{x^{2}+5 x-1}{sqrt{x}} d x ) | 12 |
| 1439 | Evaluate: ( int frac{2 x^{2}+1}{x^{2}left(x^{2}+4right)} d x ) | 12 |
| 1440 | The value of ( int_{0}^{pi / 2} frac{x sin x cos x}{sin ^{4} x+cos ^{4} x} d x ) is A ( cdot pi^{2} / 8 ) B . ( pi^{2} / 16 ) c. ( 3 pi^{2} / 4 ) D. ( pi^{2} / 2 ) | 12 |
| 1441 | Evaluate the given integral. ( int x cdot sin ^{-1} x ) | 12 |
| 1442 | ( intleft(frac{cos ^{3} x+cos ^{5} x}{sin ^{2} x+sin ^{4} x}right) d x= ) A ( cdot sin x-frac{2}{sin x}-6 tan ^{-1}(sin x)+c ) в. c. ( sin x+frac{2}{sin x}-6 tan ^{-1}(sin x)+c ) D. ( sin x+frac{2}{sin x}+6 tan ^{-1}(sin x)+c ) sin ( x+6 tan ^{-1}(sin x)+c ) | 12 |
| 1443 | If ( int_{-2}^{3} f(x) d x=5 ) and ( int_{1}^{3}{2-f(x)} d x=6 ) then the value of ( int_{-2}^{1} f(x) d x ) is? A . -5 B. 3 c. -7 D. – | 12 |
| 1444 | Evaluate ( int frac{x^{5} d x}{sqrt{left(1+x^{3}right)}}= ) A. ( frac{2}{3} sqrt{left(1+x^{3}right)+left(x^{2}+2right)} ) B. ( frac{2}{9} sqrt{left(1+x^{3}right)-left(x^{3}-4right)} ) c. ( frac{2}{9} sqrt{left(1+x^{3}right)left(x^{3}+4right)} ) D. ( frac{2}{3} sqrt{frac{left(1+x^{3}right)^{3}}{3}-left(1+x^{3}right)} ) | 12 |
| 1445 | ( int cos (log x) d x=ldots ldots ldots ldots quad+quad c ) A. ( frac{x}{2}[cos (log x)+sin (log x)] ) B ( cdot frac{x}{4}[cos (log x)+sin (log x)] ) c. ( frac{x}{2}[cos (log x)-sin (log x)] ) D. ( frac{x}{2}[sin (log x)+cos (log x)] ) | 12 |
| 1446 | The value of D.I. ( int_{-2010}^{2010} x^{2010} cot ^{-1}(2010 x) d x ) is equal to A ( cdot frac{pi}{2011}(2011)^{2010} ) B. ( frac{pi}{2010}(2010)^{2011} ) C. ( frac{pi}{2011}(2010)^{2011} ) D. ( frac{pi}{2010}(2011)^{2010} ) | 12 |
| 1447 | If ( int_{a}^{b} x^{3} d x=0 ) and ( int_{a}^{b} x^{2} d x=frac{2}{3}, ) then what are the values of ( a ) and ( b ) respectively? A. -1,1 в. 1,1 c. ( 0, ) D. 2,-2 | 12 |
| 1448 | ( operatorname{Let} f(x)=3 x^{2} cdot sin frac{1}{x}- ) ( boldsymbol{x} cos frac{1}{boldsymbol{x}}, boldsymbol{x} neq mathbf{0}, boldsymbol{f}(mathbf{0})=mathbf{0} boldsymbol{f}left(frac{mathbf{1}}{boldsymbol{pi}}right)=mathbf{0} ) then which of the following is/are not correct. This question has multiple correct options A. ( f(x) ) is continuous at ( x=0 ) B. ( f(x) ) is non-differentiable at ( x=0 ) c. ( f(x) ) is discontinuous at ( x=0 ) D. ( f(x) ) is differentiable at ( x=0 ) | 12 |
| 1449 | A function ( f(X) ) which satisfies the relation ( boldsymbol{f}(boldsymbol{X})=boldsymbol{e}^{boldsymbol{x}}+int_{0}^{1} boldsymbol{e}^{boldsymbol{x}} boldsymbol{f}(boldsymbol{t}) boldsymbol{d} boldsymbol{t}, ) then ( boldsymbol{f}(boldsymbol{X}) ) is A ( cdot frac{e^{x}}{2-e} ) B ( cdot(e-2) e^{x} ) ( c cdot 2 e^{x} ) D. ( frac{e^{x}}{2} ) | 12 |
| 1450 | Evaluate the definite integral ( int_{0}^{frac{pi}{2}} cos ^{2} x d x ) | 12 |
| 1451 | ( operatorname{Let} f(x)=max left{3, x^{2}, frac{1}{x^{2}}right} ) for ( frac{1}{2} leq ) ( x leq 2 . ) Then the value of the integral ( int_{1 / 2}^{2} f(x) d x ) is? A ( cdot frac{11}{3} ) в. ( frac{13}{3} ) c. ( frac{14}{3} ) D. ( frac{16}{3} ) | 12 |
| 1452 | The value of ( int_{0}^{frac{pi}{4}} tan ^{2} theta d theta= ) A ( cdot frac{pi}{4}-1 ) B. ( frac{pi}{4} ) ( c cdot 1-frac{pi}{4} ) D. none of these | 12 |
| 1453 | ( int frac{(x+3) e^{x}}{(x+4)^{2}} d x= ) | 12 |
| 1454 | Solve: ( int_{0}^{pi / 6} frac{cos 2 x}{(cos x-sin x)^{2}} d x ) ( mathbf{A} cdot_{-log }left(frac{sqrt{3}-1}{2}right) ) ( ^{mathbf{B}}-log left(frac{sqrt{3}+1}{2}right) ) ( mathbf{c} cdot log left(frac{sqrt{3}+1}{2}right) ) D. None of these | 12 |
| 1455 | 17. 117 – 12°dt, = {2de, 15 = 2* dr and 1a = $ 2 1 (b) 11 > 12 (©) 13 = 14 (d) 13>I, | 12 |
| 1456 | Solve: ( int frac{5 x-2}{1+2 x+3 x^{2}} cdot d x ) | 12 |
| 1457 | ( int frac{3 a x}{b^{2}+c^{2} x^{2}} d x ) | 12 |
| 1458 | ( int(x+2) sqrt{3 x+5} d x ) | 12 |
| 1459 | If ( boldsymbol{f}(boldsymbol{y})=boldsymbol{e}^{boldsymbol{y}}, boldsymbol{g}(boldsymbol{y})=boldsymbol{y} ; boldsymbol{y}>mathbf{0} ) and ( boldsymbol{F}(boldsymbol{t})=int_{0}^{t} boldsymbol{f}(boldsymbol{t}-boldsymbol{y}) boldsymbol{g}(boldsymbol{y}) boldsymbol{d} boldsymbol{y}, ) then which of the following is true? A ( cdot F(t)=e^{t}-(1+t) ) B ( cdot F(t)=t e^{t} ) c. ( F(t)=t e^{-t} ) D. ( F(t)=1-e^{-t}(1+t) ) | 12 |
| 1460 | Evaluate ( int_{-1}^{1} log frac{2-x}{2+x} d x ) | 12 |
| 1461 | Find ( int a^{x} cdot e^{x} d x ) | 12 |
| 1462 | Solve ( int_{0}^{1}|boldsymbol{x}| boldsymbol{d} boldsymbol{x} ) | 12 |
| 1463 | ( int e^{a x} cdot sin (b x+c) d x ) | 12 |
| 1464 | If ( boldsymbol{f}(boldsymbol{y})=boldsymbol{e}^{boldsymbol{y}}, boldsymbol{g}(boldsymbol{y})=boldsymbol{y} ; boldsymbol{y}>mathbf{0} ) and ( boldsymbol{F}(boldsymbol{t})=int_{0}^{t} boldsymbol{f}(boldsymbol{t}-boldsymbol{y}) boldsymbol{g}(boldsymbol{y}) boldsymbol{d} boldsymbol{y}, ) then which of the following is true? A ( cdot F(t)=e^{t}-(1+t) ) B ( cdot F(t)=t e^{t} ) c. ( F(t)=t e^{-t} ) D. ( F(t)=1-e^{-t}(1+t) ) | 12 |
| 1465 | If ( int frac{(x+1)}{xleft(1+x e^{x}right)^{2}} d x=log |1-f(x)|+ ) ( f(x)+C, ) then ( f(x)= ) A. ( frac{1}{x+e^{x}} ) B. ( frac{1}{1+x e^{x}} ) c. ( frac{1}{left(1+x e^{x}right)^{2}} ) D. ( frac{1}{left(x+e^{x}right)^{2}} ) | 12 |
| 1466 | Find: ( int_{-pi}^{pi} frac{cos ^{2} x d x}{1+a^{x}} ) where ( a>0 ) | 12 |
| 1467 | Solve: ( int frac{cos x}{x} d x ) | 12 |
| 1468 | 44. The integral log x2 2 log x² + log(36–12x+x2) dx is equal to : [JEE M 2015] (a) 1 (6) 6 (C) 2 (d) 4 | 12 |
| 1469 | 10. Given a function f(x) such that (1984 – 4 Marks) (1) it is integrable over every interval on the real line and (ü) f(t+x)=f(x), for every x and a real t, then show that a+t the integralſ f (x) dx is independent of a . | 12 |
| 1470 | ( int frac{d x}{sqrt{2 a x^{3}}} ) | 12 |
| 1471 | 1/2 52. The value of sin3 x – dx is: sin x + COS X JJEE M 2019-9 April (M) T-2 (6) T-1 4 8 T-2 | 12 |
| 1472 | If ( frac{mathbf{x}^{4}+mathbf{2 4 x}^{2}+mathbf{2 8}}{left(mathbf{x}^{mathbf{2}}+mathbf{1}right)^{mathbf{3}}} ) ( =frac{mathbf{A} mathbf{x}+mathbf{B}}{mathbf{x}^{2}+mathbf{1}}+frac{mathbf{C} mathbf{x}+mathbf{D}}{left(mathbf{x}^{2}+mathbf{1}right)^{2}}+frac{mathbf{E} mathbf{x}+mathbf{F}}{left(mathbf{x}^{2}+mathbf{1}right)^{3}} ) then ( boldsymbol{A}= ) ( A ) B. – c. ( D ) | 12 |
| 1473 | Integrate the function ( frac{x+2}{sqrt{x^{2}-1}} ) | 12 |
| 1474 | Evaluate ( int_{pi / 6}^{pi / 3} frac{d x}{1+sqrt{tan x}} ) A. ( frac{pi}{12} ) в. ( frac{7 pi}{12} ) c. ( frac{5 pi}{12} ) D. None of these | 12 |
| 1475 | Integrate the rational function ( frac{boldsymbol{x}}{left(boldsymbol{x}^{2}+mathbf{1}right)(boldsymbol{x}-mathbf{1})} ) | 12 |
| 1476 | ( int frac{e^{x}(x-1)}{x^{2}} d x= ) A ( cdot frac{1}{x} e^{x}+c ) B. ( x e^{-x}+c ) c. ( frac{1}{x^{2}} e^{x}+c ) D. ( left(x-frac{1}{x}right) e^{x}+c ) | 12 |
| 1477 | ( int frac{d x}{sin x-cos x+sqrt{2}} ) is equal to. A ( cdot-frac{1}{sqrt{2}} tan left(frac{x}{2}+frac{pi}{8}right)+C ) B. ( frac{1}{sqrt{2}} tan left(frac{x}{2}+frac{pi}{8}right)+C ) c. ( frac{1}{sqrt{2}} cot left(frac{x}{2}+frac{pi}{8}right)+C ) D. ( -frac{1}{sqrt{2}} cot left(frac{x}{2}+frac{pi}{8}right)+C ) | 12 |
| 1478 | 12. If for a real number y, [y] is the greatest integer less 370/2 equal to y, then the value of the integral [2sin x] at 1/2 (a) -1 (b) 0 (1999 – 2 Marks) (c) -a/2 (d) 1/2 | 12 |
| 1479 | Evaluate: ( int_{0}^{frac{pi}{2}} cos ^{2} x d x ) | 12 |
| 1480 | ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x ) | 12 |
| 1481 | If ( f(x)=int_{-1}^{1} frac{sin x}{1+t^{2}} d t ) then ( f^{prime}left(frac{pi}{3}right) ) is A. nonexistent в. ( pi / 4 ) c. ( pi sqrt{3 / 4} ) D. none of these | 12 |
| 1482 | Evaluate: ( int frac{x+3}{(x-1)left(x^{2}+1right)} d x ) | 12 |
| 1483 | Integrate ( frac{(x+12)}{(x+1)^{2}(x-2)} ) | 12 |
| 1484 | Solve: ( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{(boldsymbol{x}+mathbf{1}) sqrt{mathbf{1}-boldsymbol{x}^{2}}} ) | 12 |
| 1485 | Integrate ( int_{0}^{2} log x d x ) | 12 |
| 1486 | ( int frac{1}{x sqrt{1-x^{3}}} d x= ) ( a log left|frac{sqrt{1-x^{3}}-1}{sqrt{1-x^{3}}+1}right|+b, ) then a is equal to A ( cdot frac{1}{3} ) B. ( frac{2}{3} ) ( c cdot frac{-1}{3} ) D. ( frac{-2}{3} ) | 12 |
| 1487 | ( int sin ^{8} x cos x d x ) | 12 |
| 1488 | ( int frac{d x}{(x-1) sqrt{x^{2}-1}}= ) A ( -sqrt{frac{x-1}{x+1}}+C ) в. ( sqrt{frac{x-1}{x^{2}+1}}+C ) c. ( -sqrt{frac{x+1}{x-1}}+C ) D. ( sqrt{frac{x^{2}+1}{x-1}}+C ) | 12 |
| 1489 | Evaluate ( int_{-sqrt{2}}^{2 pi} frac{2 x^{7}+3 x^{6}-10 x^{5}-7 x^{3}-12 x^{2}+}{x^{2}+2} ) ( ^{mathrm{A}} cdot frac{pi}{2 sqrt{2}}+frac{16 sqrt{2}}{5} ) B. ( frac{pi}{4 sqrt{2}}-frac{8 sqrt{2}}{5} ) c. ( frac{pi}{4 sqrt{2}}+frac{8 sqrt{2}}{5} ) D. ( frac{pi}{2 sqrt{2}}-frac{16 sqrt{2}}{5} ) | 12 |
| 1490 | 4. 3/2 Find the value of 1 |x sin at x | dx | 12 |
| 1491 | Set of values of ( x ) in [0,7] for which ( g(x) ) is negative is ( A cdot(2,7) ) B. (3,7) ( c cdot(4,6) ) D. (5,7) | 12 |
| 1492 | Evaluate: ( int_{-1}^{1} sin ^{5} x cos ^{4} x d x ) | 12 |
| 1493 | Evaluate the following integrals: ( int sqrt{2 x-x^{2}} d x ) A ( cdot(x-1) sqrt{2 x-x^{2}}+frac{1}{2} sin ^{-1}(x-1)+C ) B. ( frac{1}{2}(x-1) sqrt{2 x-x^{2}}+2 sin ^{-1}(x-1)+C ) c. ( frac{1}{2}(x-1) sqrt{2 x-x^{2}}+frac{1}{2} sin ^{-1}(x-1)+C ) D. none of these | 12 |
| 1494 | Solve: ( int sqrt{(x-2)(x-2)} d x ) | 12 |
| 1495 | Evaluate: ( int frac{1}{4+9 x^{2}} d x ) | 12 |
| 1496 | ( boldsymbol{n} stackrel{L t}{rightarrow} inftyleft[frac{boldsymbol{n}+mathbf{1}}{boldsymbol{n}^{2}+mathbf{1}^{2}}+frac{boldsymbol{n}+boldsymbol{2}}{boldsymbol{n}^{2}+mathbf{2}^{2}}+ldots+right. ) ( left.frac{boldsymbol{n}+boldsymbol{n}}{boldsymbol{n}^{2}+boldsymbol{n}^{2}}right]= ) A. ( frac{pi}{4}+frac{1}{2} log 2 ) B. ( frac{pi}{4}-frac{1}{2} log 2 ) c. ( frac{pi}{2}+frac{1}{2} log 2 ) D. ( frac{pi}{4}+frac{1}{4} log 2 ) | 12 |
| 1497 | Evaluate the definite integral ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{mathbf{1}+boldsymbol{x}}-sqrt{boldsymbol{x}}} ) | 12 |
| 1498 | If the function ( boldsymbol{f}:[mathbf{0}, mathbf{8}] rightarrow boldsymbol{R} ) is differentiable and ( 0<alpha<1<beta<2 ) then ( int_{0}^{8} f(t) d t ) is equal to? A ( cdot 3left[a^{3} fleft(a^{2}right)+beta^{2} fleft(beta^{2}right)right] ) B・3 ( left[a^{3} f(a)+beta^{3} f(beta)right. ) c ( cdot 3left[a^{2} fleft(a^{2}right)+beta^{2} fleft(beta^{3}right)right] ) D ( cdot 3left[a^{2} fleft(a^{3}right)+beta^{2} fleft(beta^{3}right)right] ) | 12 |
| 1499 | If ( frac{boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1}}{boldsymbol{x}^{2}+mathbf{2} boldsymbol{x}+mathbf{1}}=boldsymbol{A}+frac{boldsymbol{B}}{boldsymbol{x}+mathbf{1}}+ ) ( frac{C}{(x+1)^{2}} ) then ( A-B= ) A . ( 4 C ) B. ( 4 C+1 ) ( c .3 C ) D. ( 2 C ) | 12 |
| 1500 | Evaluate ( int_{0}^{1} frac{1}{sqrt{1+x}-sqrt{x}} d x ) | 12 |
| 1501 | If ( int 2^{2 x} cdot 2^{x} d x=A cdot 2^{2^{x}}+c, ) then ( A=? ) A ( cdot frac{1}{log 2} ) в. ( log 2 ) c. ( (log 2)^{2} ) D. ( frac{1}{(log 2)^{2}} ) | 12 |
| 1502 | ( int frac{1}{9 x^{2}-25} d x= ) A ( cdot frac{1}{30} log mid frac{3 x+5}{3 x-5} ) B cdot ( log |x+sqrt{3 x-5}| ) c. ( frac{1}{30} log left|frac{3 x-5}{3 x+5}right| ) D cdot ( log mid x-sqrt{3 x-5} ) | 12 |
| 1503 | The value of ( int_{-4}^{-5} e^{(x+5)^{2}} d x+ ) ( mathbf{3} int_{1 / 3}^{2 / 3} e^{9(x-2 / 3)^{2}} d x ) A . ( 2 / 5 ) в. ( 1 / 5 ) ( c cdot 1 / 2 ) D. none of these | 12 |
| 1504 | Integrate ( frac{tan ^{4} sqrt{x}+sec ^{2} sqrt{x}}{sqrt{x}} ) The solution is ( frac{2 tan ^{3}(sqrt{x})}{m}- ) ( 2 tan sqrt{x}+2 sqrt{x}+2 tan sqrt{x}+C . ) Find ( m ) | 12 |
| 1505 | Solve ( int sqrt{frac{boldsymbol{a}-boldsymbol{x}}{boldsymbol{a}+boldsymbol{x}}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1506 | ( int_{pi / 4}^{3 pi / 4} frac{d x}{1+cos x} ) is equal to: A .2 B. -2 ( c cdot frac{1}{2} ) D. ( -frac{1}{2} ) | 12 |
| 1507 | ( (cos (log x) d x= ) ( mathbf{A} cdot x[cos (log x)-sin (log x)]+c ) B. ( frac{x}{2}[cos (log x)-sin (log x)]+c ) c. ( frac{log x}{2}[cos x+sin x]+c ) D ( cdot frac{x}{2}[cos (log x)+sin (log x)]+c ) | 12 |
| 1508 | Prove that: ( int frac{x^{2}}{1+x^{3}} d x ) | 12 |
| 1509 | ( frac{boldsymbol{x}^{boldsymbol{4}}+boldsymbol{2} boldsymbol{4} boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{8}}{left(boldsymbol{x}^{2}+mathbf{1}right)^{3}}= ) A. ( frac{1}{x^{2}+1}+frac{22}{left(x^{2}+1right)^{2}}+frac{5}{left(x^{2}+1right)^{3}} ) B. ( frac{1}{x^{2}+1}-frac{22}{left(x^{2}+1right)^{2}}+frac{5}{left(x^{2}+1right)^{3}} ) C ( frac{1}{x^{2}+1}+frac{22}{left(x^{2}+1right)^{2}}+frac{28}{left(x^{2}+1right)^{3}} ) D. ( frac{1}{x^{2}+1}+frac{23}{left(x^{2}+1right)^{2}}+frac{4}{left(x^{2}+1right)^{3}} ) | 12 |
| 1510 | ( int frac{t^{2}+t}{t} d t ) | 12 |
| 1511 | Evaluate: ( int x log (x+1) d x ) | 12 |
| 1512 | Obtain: ( int frac{(3 x+2)}{(x+1)(x+2)(x-3)} d x ) | 12 |
| 1513 | If ( I=int_{8}^{15} frac{d x}{(x-3) sqrt{x+1}} ) then ( I ) equals A ( cdot frac{1}{2} log frac{5}{3} ) B. ( 2 log frac{1}{3} ) c. ( frac{1}{2}-log frac{1}{5} ) D. ( 2 log frac{5}{3} ) | 12 |
| 1514 | ff ( f(x) ) a polynomial of degree 2 in ( x ) such that ( boldsymbol{f}(mathbf{0})=boldsymbol{f}(mathbf{1})=mathbf{3} boldsymbol{f}(mathbf{2})=-mathbf{3} ) then ( int frac{boldsymbol{f}(boldsymbol{x})}{boldsymbol{x}^{3}-mathbf{1}} boldsymbol{d} boldsymbol{x}= ) A ( cdot log left|x^{2}+x+1right|+log |x+1|+c ) B ( cdot log |x-1|+frac{2}{sqrt{3}} tan ^{-1}left(frac{2 x+1}{sqrt{3}}right)+c ) ( ^{mathbf{c}} cdot log left|x^{2}+x+1right|+frac{2}{sqrt{3}} tan ^{-1}left(frac{2 x+1}{sqrt{3}}right)+c ) D ( cdot log left|x^{2}+x+1right|-log |x-1|+frac{2}{sqrt{3}} tan ^{-1}left(frac{2 x+1}{sqrt{3}}right)+c ) | 12 |
| 1515 | 12 (log x-1)! dx is equal to 120 1+(log x)?). log x +C (a) (log x)2 +1 xet +C c a ) dog op die 1+x | 12 |
| 1516 | ( frac{(x+1)^{2}}{x^{3}+x}=frac{A}{x}+frac{B x+C}{x^{2}+1} Rightarrow ) ( sin ^{-1}left[frac{A}{C}right]= ) A ( cdot frac{pi}{6} ) в. ( frac{pi}{4} ) ( c cdot frac{pi}{3} ) ( D cdot frac{pi}{2} ) | 12 |
| 1517 | A function ( f ) is defined by ( f(x)= ) ( frac{1}{2^{r-1}}, frac{1}{2 r}<x leq frac{1}{2^{r-1}}, r=1,2,3, dots . ) then the value of ( int_{0}^{1} f(x) d x ) is equal A B. ( c cdot frac{2}{3} ) D. | 12 |
| 1518 | Evaluate the given definite integrals as limit of sums: ( int_{0}^{4}left(x+e^{2 x}right) d x ) | 12 |
| 1519 | Find ( int frac{x^{4}}{(x-1)left(x^{2}+1right)} d x ) | 12 |
| 1520 | Solve : ( int frac{u}{v} d x ) | 12 |
| 1521 | Evaluate ( int frac{boldsymbol{x}^{2}}{mathbf{9}+mathbf{1 6 x}^{mathbf{6}}} boldsymbol{d} boldsymbol{x} ) A ( frac{1}{16} tan ^{-1}left(frac{4 x^{3}}{3}right)+c ) в. ( frac{1}{36} tan ^{-1}left(frac{3 x^{3}}{4}right)+c ) c. ( frac{1}{16} tan ^{-1}left(frac{3 x^{3}}{4}right)+c ) D. ( frac{1}{36} tan ^{-1}left(frac{4 x^{3}}{3}right)+c ) | 12 |
| 1522 | ( int a^{m x} b^{n x} d x ) | 12 |
| 1523 | Evaluate: ( int sqrt{x^{2}} mathrm{d} x ) | 12 |
| 1524 | ( int frac{x^{4}}{(x-1)left(x^{2}+1right)} d x ) (Assuming all conditions for the domain to be met) A ( x^{2}+x+frac{1}{2} log (x-1)-frac{1}{4} log left(x^{2}+1right)-frac{1}{2} tan ^{-1} x+c ) B ( cdot frac{1}{2} x^{2}+x+frac{1}{2} log (x-1)-frac{1}{4} log left(x^{2}+1right)-frac{1}{2} tan ^{-1} x+c ) C ( cdot frac{1}{2} x^{2}+x-frac{1}{2} log (x-1)-frac{1}{4} log left(x^{2}+1right)-frac{1}{2} tan ^{-1} x+c ) D ( cdot frac{1}{2} x^{2}-x+frac{1}{2} log (x-1)-frac{1}{4} log left(x^{2}+1right)+frac{1}{2} tan ^{-1} x+c ) | 12 |
| 1525 | Solve ( int_{0}^{h} x(h-x) d x ) ( ^{A} cdot_{I}=frac{h^{3}}{3} ) в. ( _{I=} frac{h^{3}}{6} ) ( ^{mathrm{C}} cdot_{I}=-frac{h^{3}}{6} ) D. None of these | 12 |
| 1526 | ( int frac{1}{sqrt{4+x^{2}}} d x ) | 12 |
| 1527 | ( int frac{x^{3}-1}{x^{3}+x} d x ) equal to ( mathbf{A} cdot x-log x+log left(x^{2}+1right)-tan ^{-1} x+c ) B. ( x-log x+frac{1}{2} log left(x^{2}+1right)-tan ^{-1} x+c ) c. ( x+log x+frac{1}{2} log left(x^{2}+1right)+tan ^{-1} x+c ) D. ( x+log x-frac{1}{2} log left(x^{2}+1right)-tan ^{-1} x+c ) | 12 |
| 1528 | Evaluate : ( int frac{(4 x+1) d x}{x^{2}+3 x+2} ) A. [ begin{array}{l}text { s. } \ begin{array}{l}text { s. } 2 log left|x^{2}+5 x+2right|-5 log left|frac{x+1}{x+2}right|+C \ =2 log left|x^{2}+3 x+2right|-5 log left|frac{x+3}{x+2}right|+Cend{array} \ text { 0. }=2 log left|x^{2}+3 x+2right|-5 log left|frac{x+1}{x+2}right|+Cend{array} ] | 12 |
| 1529 | If ( I_{n}=int(log x)^{n} d x, ) then ( I_{6}+6 I_{5}= ) ( mathbf{A} cdot x(log x)^{5} ) B. ( -x(log x)^{5} ) ( mathbf{c} cdot x(log x)^{6} ) ( mathbf{D} cdot-x(log x)^{6} ) | 12 |
| 1530 | ( int sin ^{2 / 3} x cos ^{3} x d x ) | 12 |
| 1531 | If ( int e^{x}(operatorname{nn} x+x operatorname{cn} x+1) d x=f(x)+ ) cwhenf( (1)=0, ) then ( f(e) ) is equal to ( A ) B ( cdot e^{s} ) ( mathbf{c} cdot e^{s-1} ) D. ( e^{e+1} ) | 12 |
| 1532 | ff ( f(x)left|begin{array}{cccc}x & & cos x & e^{x^{2}} \ sin & x & x^{2} & sec x \ tan x & x^{4} & 2 x^{2}end{array}right| ) then ( int_{-pi / 2}^{pi / 2} boldsymbol{f}(boldsymbol{x}) boldsymbol{d} boldsymbol{x}= ) A . B. ( c cdot 2 ) ( D ) | 12 |
| 1533 | Suppose ( boldsymbol{f}:[mathbf{0}, boldsymbol{pi}] rightarrow mathbb{R} ) satisfied ( boldsymbol{f}(boldsymbol{x})+ ) ( boldsymbol{f}(boldsymbol{pi}-boldsymbol{x})=1 ) for all ( boldsymbol{x} . ) Then ( int_{0}^{pi} f(x) sin x d x ) is A ( cdot frac{1}{4} ) B. ( frac{1}{2} ) ( c cdot frac{3}{4} ) D. | 12 |
| 1534 | If ( int frac{operatorname{cosec}^{2} x-2010}{cos ^{2010 x}} d x= ) ( -frac{boldsymbol{f}(boldsymbol{x})}{(boldsymbol{g}(boldsymbol{x}))^{2010}}+C ; ) then the number of solutions where equation ( frac{f(x)}{g(x)}={x} ) in ( [mathbf{0}, mathbf{2} boldsymbol{pi}] ) is / are: ( A cdot 0 ) B. 1 c. 2 D. 3 | 12 |
| 1535 | Prove that ( : int frac{1}{a^{2}-x^{2}} d x= ) ( frac{1}{2 a} ln left|frac{a+x}{a-x}right|+c ) | 12 |
| 1536 | Assertion The equation ( 4 x^{3}-9 x^{2}+2 x+1=0 ) has atleast one real root in (0,1) Reason If ‘f’ is a continuous function such that ( int_{a}^{b} f(x)=0, ) the the equation ( f(x)=0 ) has atleast one real root in ( (a, b) ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1537 | Find: ( int frac{left(x^{4}-xright)^{frac{1}{4}}}{x^{5}} d x ) | 12 |
| 1538 | Evaluate the integral ( int_{0}^{pi / 2} frac{cos x}{1+sin ^{2} x} d x ) ( A ) в. ( pi / 3 ) c. ( pi / 2 ) D. ( pi / 4 ) | 12 |
| 1539 | The value of ( int e^{2 x}left(frac{1}{x}-frac{1}{2 x^{2}}right) d x ) is ( ^{A} cdot frac{e^{2 x}}{2}+c ) B. ( frac{e^{2 x}}{2 x}+c ) c. ( frac{e^{2 x}}{3 x}+c ) D. ( frac{e^{2 x}}{x}+c ) | 12 |
| 1540 | Evaluate ( int_{0}^{1} e^{2-3 x} d x ) as a limit of a sum | 12 |
| 1541 | Evaluate the following integral: ( int_{0}^{2}left(x^{2}+3right) d x ) | 12 |
| 1542 | sinx 49. The value of Bica undx is : (JEEM 1+ (b) 47 (C) | 12 |
| 1543 | Evaluate the given integral. ( int sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x ) | 12 |
| 1544 | Evaluate: ( int sqrt[9]{x^{-8}} d x ) A ( cdot 9 x^{frac{1}{9}}+c ) ( c ) B ( .9 x^{8}+c ) c. ( x^{frac{1}{9}}+c ) D. ( x^{frac{8}{9}}+c ) | 12 |
| 1545 | ( int_{1}^{2}left(x+frac{1}{x}right)^{3 / 2} frac{x^{2}-1}{x^{2}} d x ) ( mathbf{A} cdot frac{5}{2} sqrt{left(frac{5}{2}right)}+frac{8}{5} sqrt{2} ) B ( cdot frac{5}{2} sqrt{left(frac{5}{2}right)}-frac{8}{5} sqrt{2} ) ( mathrm{C} cdot sqrt{left(frac{5}{2}right)}-frac{8}{5} sqrt{2} ) D ( frac{3}{2} sqrt{left(frac{3}{2}right)}-frac{8}{5} sqrt{2} ) | 12 |
| 1546 | ( int_{2}^{5} sqrt{frac{5-x}{x-2}} d_{X}= ) ( A ) B . ( pi / 2 ) ( mathrm{c} .3 pi / 2 ) D. ( pi / ) | 12 |
| 1547 | Evaluate ( int_{0}^{2}left(x^{2}-3 x+2right) d x ) | 12 |
| 1548 | Solve ( int 2 x^{3} e^{x^{2}} ) A ( cdotleft(x^{2} e^{x^{2}}-e^{x^{2}}right)+C ) B. ( frac{1}{2}left(x^{2} e^{x^{2}}-e^{x^{2}}right)+C ) C ( cdot frac{1}{2}left(x^{2} e^{x^{2}}+e^{x^{2}}right)+C ) D ( cdot frac{1}{4}left(x^{2} e^{x^{2}}+e^{x^{2}}right)+C ) | 12 |
| 1549 | ( int sin theta cos theta d theta ) | 12 |
| 1550 | Evaluate the following integral: ( int sec ^{2} x d x ) ( A cdot 2 tan x+C ) B. ( tan 2 x+C ) ( mathbf{c} cdot tan x+C ) D. None of these | 12 |
| 1551 | Solve ( :left(x^{2}-1right) frac{d y}{d x}+2 x y=frac{1}{x^{2}-1} ) | 12 |
| 1552 | The value of ( int_{1}^{2}[boldsymbol{f}{boldsymbol{g}(boldsymbol{x})}]^{-1} cdot boldsymbol{f}^{prime}{boldsymbol{g}(boldsymbol{x})} cdot boldsymbol{g}^{prime}(boldsymbol{x}) boldsymbol{d} boldsymbol{x} ) where ( g(1)=g(2), ) is equal to? A . 1 B . 2 c. 0 D. none of these | 12 |
| 1553 | Evaluate ( int_{4}^{12} x d x ) | 12 |
| 1554 | Find an anti derivative (or integral) of the given function by the method of inspection. ( e^{2 x} ) | 12 |
| 1555 | If ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{f}(boldsymbol{x})=mathbf{4} boldsymbol{x}^{3}-frac{boldsymbol{3}}{boldsymbol{x}^{4}} ) such that ( boldsymbol{f}(2)=0 . ) Then ( boldsymbol{f}(boldsymbol{x}) ) is A ( cdot x^{4}+frac{1}{x^{3}}-frac{129}{8} ) B. ( x^{3}+frac{1}{x^{4}}+frac{129}{8} ) c. ( x^{4}+frac{1}{x^{3}}+frac{129}{8} ) D. ( x^{3}+frac{1}{x^{4}}-frac{129}{8} ) | 12 |
| 1556 | Evaluate the definite integral ( int_{0}^{frac{pi}{4}} tan x d x ) | 12 |
| 1557 | The value of integral ( int_{pi / 4}^{3 pi / 4} frac{x}{1+sin x} d x ) is : A ( cdot pi sqrt{2} ) B . ( frac{pi}{2}(sqrt{2}+1) ) c. ( pi(sqrt{2}-1) ) D. ( 2 pi(sqrt{2}-1) ) | 12 |
| 1558 | Evaluate: ( int_{0}^{1} x^{3}+3 x^{2} d x ) | 12 |
| 1559 | Evaluate : ( int frac{boldsymbol{d} boldsymbol{x}}{mathbf{3} sin ^{2} boldsymbol{x}+sin boldsymbol{x}} cos boldsymbol{x}+mathbf{1} ) A ( cdot frac{2}{sqrt{15}} tan ^{-1}left(frac{2(cot x-2)}{sqrt{15}}right) ) в. ( frac{2}{sqrt{3}} tan ^{-1}left(frac{3 tan x-1}{sqrt{15}}right)+c ) c. ( frac{2}{sqrt{3}} tan ^{-1}left(frac{3 tan x+1}{sqrt{15}}right)+c ) D. ( frac{2}{sqrt{15}} tan ^{-1}left(frac{2(cot x-2)}{sqrt{15}}right) ) | 12 |
| 1560 | Simplify:( int x ell n sqrt{x} d x ) | 12 |
| 1561 | If ( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{boldsymbol{x}-sqrt{mathbf{9} boldsymbol{x}^{2}+mathbf{4} boldsymbol{x}+mathbf{6}}} ) to evaluate ( I ), one of the most proper substitution could be A. ( sqrt{9 x^{2}+4 x+6}=u pm 3 x ) B. ( sqrt{9 x^{2}+4 x+6}=3 u pm x ) c. ( _{x}=frac{1}{t} ) D. ( 9 x^{2}+4 x+6=frac{1}{t} ) | 12 |
| 1562 | 19. (1990 – 4 Marks Prove that for any positive integer k, sin 2kx = 2[cos x + cos 3x + ………….. +cos (2k-1)x] sin x T/2 Hence prove that sin 2kx cot x dx = | 12 |
| 1563 | Evaluate the following integrals: ( int frac{cos x}{sqrt{4+sin ^{2} x}} d x ) | 12 |
| 1564 | ( int sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x ) is equal to A ( cdot 2left(x tan ^{-1} x+ln left|cos left(tan ^{-1} xright)right|right)+C ) B・ ( 2left[left(x tan ^{-1} xright)^{2}+ln left|sec left(tan ^{-1} xright)right|right]+c ) c. ( 2left[left(x tan ^{-1} xright)^{2}-ln left|cos left(tan ^{-1} xright)right|right]+c ) D. None of these | 12 |
| 1565 | Evaluate ( int_{-pi}^{pi} frac{2 x(1+sin x)}{1+cos ^{2} x} d x ) | 12 |
| 1566 | 5. Letf:(0,0) Rand F Sdt. If F(x) = x²(1+x), then (4) equals (a) 5/4 (6) 7 (c) 4 (20015) (d) 2. I . | 12 |
| 1567 | nit+V chw that 27. Show that sin x dx = 2n +1-COS v where n is a positive integer and 0 Sy<TT. (1994 – 4 Marks | 12 |
| 1568 | ( int(tan x-cot x)^{2} d x= ) A ( cdot tan x+x+c ) B. ( tan x-x+c ) c. ( tan x-cot x+c ) D. ( tan x-cot x-4 x+c ) | 12 |
| 1569 | Number of Partial Fractions of ( frac{x^{3}+x^{2}+1}{x^{4}+x^{2}+1} ) is ( A cdot 2 ) B. 3 ( c cdot 4 ) D. | 12 |
| 1570 | Prove ( int frac{d x}{sqrt{a^{2}-x^{2}}} ) | 12 |
| 1571 | ( int_{1}^{3} log x d x=ldots ldotsleft(x>0 in R^{+}right) ) A. ( -2+log 27 ) B. ( -2+log 9 ) c. ( 2+log 27 ) ( ^{mathrm{D}} cdot log left(frac{27}{e}right) ) | 12 |
| 1572 | The value of ( int_{0}^{2 pi} frac{d x}{e^{sin x}+1} ) A . ( pi ) B. ( c cdot 2 pi ) D. | 12 |
| 1573 | ( frac{2 x+1}{(x-1)left(x^{2}+1right)}=frac{A}{x-1}+ ) ( frac{B x+C}{x^{2}+1} Rightarrow C= ) A . B. 1/2 ( c cdot-1 / 2 ) D. 5/2 | 12 |
| 1574 | ( int cos ^{-1}left(frac{1}{x}right) d x ) equal to A ( cdot x sec ^{-1} x+cosh ^{-1} x+c ) B ( cdot x sec ^{-1} x-cosh ^{-1} x+c ) c. ( x sec ^{-1} x-sin ^{-1} x+c ) D. ( x sec ^{-1} x+sin ^{-1} x+c ) | 12 |
| 1575 | ntegrate the function ( frac{mathbf{5 x}}{(x+1)left(x^{2}+9right)} ) | 12 |
| 1576 | Solve it ( mathbf{2} boldsymbol{I}=int_{boldsymbol{O}}^{boldsymbol{Q}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1577 | Evaluate: ( int frac{tan x}{(cos x)^{2}} d x ) | 12 |
| 1578 | If ( int e^{x}(1+x) sec ^{2}left(x e^{x}right) d x=f(x)+c ) then ( boldsymbol{f}(boldsymbol{x})= ) A ( cdot cos left(x e^{x}right) ) B. ( sin left(x e^{x}right) ) c. ( 2 tan ^{-1}(x) ) D. ( tan left(x e^{x}right) ) | 12 |
| 1579 | The value of the integral ( int_{frac{1}{3}}^{1} frac{left(x-x^{3}right)^{frac{1}{3}}}{x^{4}} d x ) is? ( A cdot 6 ) B. ( c .3 ) D. | 12 |
| 1580 | Evaluate: ( int frac{x-x^{2}}{x^{2}-2 x-3} d x ) | 12 |
| 1581 | 210 The value of [2 Sinx) dx where [.) represents the greatest integer function is (1995) (d) -2 (a) -ST (6) – T ( 54 | 12 |
| 1582 | The value of integral ( int_{0}^{pi} frac{x^{2} sin x}{(2 x-pi)left(1+cos ^{2} xright)} d x ) is equal to ( ^{A} cdot frac{pi^{2}}{4} ) в. ( frac{pi^{2}}{2} ) c. ( frac{pi^{2}}{6} ) D. none of these | 12 |
| 1583 | ( mathbf{f} boldsymbol{I}=int sin ^{-1}left(frac{mathbf{2} boldsymbol{x}+mathbf{2}}{sqrt{mathbf{4} boldsymbol{x}^{2}+mathbf{8} boldsymbol{x}+mathbf{1 3}}}right) boldsymbol{d} boldsymbol{x}= ) ( (x+1) tan ^{-1} frac{2 x+2}{3}-frac{A}{248} log left(4 x^{2}+right. ) ( 8 x+13)+C ) then ( A ) is equal to. | 12 |
| 1584 | If ( boldsymbol{a}>mathbf{0} ) and ( int_{0}^{a}[boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(-boldsymbol{x})] boldsymbol{d} boldsymbol{x}= ) ( int_{-a}^{a} phi(x) d x ) then one of the possible values of ( phi(x) ) can be A ( cdot f(-x) ) B . ( -f(x) ) c. ( frac{1}{2} f(x) ) D. none of these | 12 |
| 1585 | umbers), the + 1, neN (the set of natural numbers) 13. For r2 # n integral 2 sin (x2 – 1) – sin 2 (x2-1) dr is equal to: “V2 sin (x² – 1)+sin 2(x2 – 1) [JEE M 2019-9 Jan (M) (a) log.sec? (22-1) +C () zlog, /sec (x2-1))+c a) loe, bee ( (where c is a constant of integration) | 12 |
| 1586 | If ( int frac{1}{left(x^{2}+4right)left(x^{2}+9right)} d x=A tan ^{-1} frac{x}{2}+ ) ( B tan ^{-1}left(frac{x}{3}right)+C ) then ( A-B= ) A ( cdot frac{1}{6} ) B. ( frac{1}{30} ) ( c cdot-frac{1}{3} ) D. ( -frac{1}{6} ) | 12 |
| 1587 | Evaluate ( int frac{x^{3}+4 x^{2}-7 x+5}{x+2} d x ) | 12 |
| 1588 | Match the column | 12 |
| 1589 | Evaluate: ( lim _{n rightarrow infty} sum_{r=0}^{n-1} frac{1}{n+r} ) ( A cdot log 2 ) B. ( 2 log 2 ) c. ( frac{1}{2} log 2 ) D. ( frac{1}{4} log 2 ) | 12 |
| 1590 | Evaluate ( int_{0}^{pi / 2} frac{d x}{2+sin 2 x} ) A ( cdot frac{2 pi}{3} ) в. ( c cdot frac{2 pi}{5} ) D. None of these | 12 |
| 1591 | Evaluate ( int x cdot e^{-x^{2}} d x ) | 12 |
| 1592 | Evaluate ( int_{0}^{2}left(x^{2}+2right) d x ) | 12 |
| 1593 | ( operatorname{Let} f(x)=7 tan ^{8} x+7 tan ^{6} x- ) ( 3 tan ^{4} x-3 tan ^{2} x ) for all ( x inleft(-frac{pi}{2}, frac{pi}{2}right) ) then the correct expression(s) is (are) This question has multiple correct options ( int_{0}^{frac{pi}{4}} x f(x) d x=frac{1}{12} ) B. ( int_{0}^{frac{pi}{4}} f(x) d x=0 ) c. ( int_{0}^{frac{pi}{4}} x f(x) d x=frac{1}{6} ) D ( int_{0}^{frac{pi}{4}} x f(x) d x=1 ) | 12 |
| 1594 | Find ( intleft(sqrt{boldsymbol{x}}+frac{mathbf{1}}{sqrt{boldsymbol{x}}}right) boldsymbol{d} boldsymbol{x} ) | 12 |
| 1595 | Find the integrals of the functions. i) ( sin ^{2}(2 x+5) ) ii) ( sin 3 x cos 4 x ) iii) ( cos 2 x cos 4 x cos 6 x ) iv) ( sin ^{3}(2 x+1) ) | 12 |
| 1596 | Integrate with respect to ( x ). ( e^{x} sin x ) | 12 |
| 1597 | Evaluate ( : int_{0}^{1} e^{2-3 x} d x ) ( mathbf{A} cdot e^{2}-e ) B. ( frac{1}{3}left(e^{2}-eright) ) c. ( frac{1}{3}left(e^{2}-frac{1}{e}right) ) D. ( frac{1}{2}left(e^{2}-frac{1}{e}right) ) | 12 |
| 1598 | ( operatorname{Let} int_{0}^{a} f(x) d x=lambda ) and ( int_{0}^{a} boldsymbol{f}(boldsymbol{2} boldsymbol{a}-boldsymbol{x}) boldsymbol{d} boldsymbol{x}=boldsymbol{mu} ) Then ( int_{0}^{2 a} f(x) d x ) is equal to? ( A cdot lambda+mu ) B. ( lambda-mu ) ( c cdot 2 lambda-mu ) D. ( lambda-2 mu ) | 12 |
| 1599 | The value of ( int_{frac{7 pi}{4}}^{frac{7 pi}{3}} sqrt{tan ^{2} x} d x ) is equal to A. ( 2 log 2 ) B. ( log 2 sqrt{2} ) ( c cdot log 2 ) D. ( log sqrt{2} ) | 12 |
| 1600 | Evalaute the integral ( int_{0}^{pi} x f(sin x) d x ) ( mathbf{A} cdot 2 pi ) ( ^{mathbf{B}} cdot pi int_{0}^{pi / 2} f(cos x) d x ) ( ^{mathbf{c}} cdot_{pi} int_{0}^{pi} f(cos x) d x ) D. ( pi int_{0}^{pi} f(sin x) d x ) | 12 |
| 1601 | Let ( boldsymbol{f}: mathbb{R} rightarrow mathbb{R} ) be a differentiable function such that ( boldsymbol{f}(mathbf{0})=mathbf{0}, boldsymbol{f}left(frac{boldsymbol{pi}}{mathbf{2}}right)=mathbf{3} ) and ( boldsymbol{f}^{prime}(mathbf{0})=mathbf{1 .} ) If ( boldsymbol{g}(boldsymbol{x})= ) ( int_{x}^{frac{pi}{2}}left[f^{prime}(t) operatorname{cosec} t-cot t operatorname{cosec} t f(t)right] d t ) for ( boldsymbol{x} inleft(mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right], ) then ( lim _{boldsymbol{x} rightarrow 0} boldsymbol{g}(boldsymbol{x})= ) | 12 |
| 1602 | Assertion If ( a>0 ) and ( b^{2}-4 a c0, b^{2}-4 a c<0, ) then ( a x^{2}+b x+ ) ( C ) can be written as sum of two squares. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1603 | T/2 36. The value of the integral Ttx x² + In 22 cos xdx is TT – X (2012) 2 +4 | 12 |
| 1604 | f ( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{(1+sin boldsymbol{x})^{4}}= ) ( frac{-boldsymbol{A}}{mathbf{4} mathbf{9} mathbf{9} mathbf{2}}left(frac{mathbf{1}}{mathbf{7}} boldsymbol{u}^{mathbf{7}}+frac{mathbf{3}}{mathbf{5}} boldsymbol{u}^{mathbf{5}}+boldsymbol{u}^{mathbf{3}}+boldsymbol{u}right)+boldsymbol{C} ) where ( u=frac{1-2 sin x}{1+sin x} ) then ( A ) is equal | 12 |
| 1605 | Assertion Let ( f: R rightarrow R ) be defined as ( f(x)= ) ( a x^{2}+b x+c, ) where ( a, b, c varepsilon R ) and ( a neq ) 0 If ( f(x)=0 ) is having non-real roots, then ( int frac{d x}{f(x)}=lambda tan ^{-1}(g(x))+k ) where ( lambda, k ) are constants and ( g(x) ) is linear function of ( x ) Reason ( tan left(tan ^{-1} g(x)right)=g(x) forall x in R ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct | 12 |
| 1606 | Solve: ( int frac{x^{3}+x+1}{x^{2}-1} d x ) A ( cdot frac{x^{2}}{2}+log left(x^{2}-1right)+frac{1}{2} log frac{x-1}{x+1}+c ) B ( cdot frac{x^{2}}{2}+log left(x^{2}-1right)-frac{1}{2} log frac{x-1}{x+1}+c ) ( ^{mathbf{C}} cdot frac{x^{2}}{2}-log left(x^{2}-1right)+frac{1}{2} log frac{x+1}{x-1}+c ) D ( cdot frac{x^{2}}{2}-log left(2 x^{2}-1right)-frac{1}{2} log frac{x-1}{x+1}+c ) | 12 |
| 1607 | Evaluate : ( int e^{x}left[frac{1+x log x}{x}right] d x ) | 12 |
| 1608 | ( int frac{1}{(x-1)(x-2)} d x ) | 12 |
| 1609 | ( int_{alpha}^{beta} sqrt{(x-alpha)(beta-x)} d x ) equals A ( cdot frac{pi}{2}(beta-alpha) ) В ( cdot frac{pi}{8}(beta-alpha) ) c. ( frac{pi}{8}(beta-alpha)^{2} ) D. None of these | 12 |
| 1610 | ( int 3 V-V^{3} d V ) | 12 |
| 1611 | f ( int_{1}^{a}left(3 x^{2}+2 x+1right) d x=11, ) find real values of a. | 12 |
| 1612 | Evaluate the definite integral: ( int_{0}^{pi / 2} cos ^{2} x d x ) | 12 |
| 1613 | Solve ( int frac{x^{2} tan ^{-1} x^{3}}{1+x^{6}} d x ) | 12 |
| 1614 | ( int frac{d x}{16 x^{2}-25} ) | 12 |
| 1615 | If ( int frac{4 e^{x}+6 e^{-x}}{9 e^{x}-4 e^{-x}} d x=A x+b ln left(9 e^{2 x}-right. ) 4) ( +C ; ) then; value of ( A, B, & C ) are A ( cdot A=-frac{3}{2}, B=frac{35}{36}, C varepsilon R ) в. ( A=frac{3}{2}, B=frac{-35}{36}, C varepsilon R ) c. ( A=-frac{3}{2}, B=frac{35}{36}, C>0 ) D. None of these | 12 |
| 1616 | ( lim _{n rightarrow infty}left[frac{1}{n^{2}} sec ^{2} frac{1}{n^{2}}+frac{2}{n^{2}} sec ^{2} frac{4}{n^{2}}+ldots+frac{1}{n}right. ) equals A ( cdot frac{1}{2} operatorname{cosec} 1 ) B ( cdot frac{1}{2} sec 1 ) c. ( frac{1}{2} tan 1 ) D. tan 1 | 12 |
| 1617 | The value of ( int frac{cos x+x sin x}{x^{2}+x cos x} d x ) is ( mathbf{A} cdot log left|frac{sin x}{1+cos x}right|+C ) ( mathbf{B} cdot log left|frac{sin x}{x+cos x}right|+C ) ( mathbf{C} cdot log left|frac{2 sin x}{x+cos x}right|+C ) ( ^{mathbf{D}} cdot log left|frac{x}{x+cos x}right|+C ) | 12 |
| 1618 | Find area of ( boldsymbol{y}=boldsymbol{x}^{2} ) from ( boldsymbol{x}=boldsymbol{2} ) to ( boldsymbol{x}=boldsymbol{4} ) | 12 |
| 1619 | ( int_{-1 / 2}^{1 / 2}(cos x)left[log left(frac{1-x}{1+x}right)right] d x ) is equal to : ( mathbf{A} cdot mathbf{0} ) B. ( c cdot e^{1 / 2} ) D. ( 2 e^{1 / 2} ) | 12 |
| 1620 | Prove that: ( int_{0}^{2 a} f(x) d x=int_{0}^{a} f(x) d x+int_{0}^{a} f(2 a- ) ( boldsymbol{x}) boldsymbol{d} boldsymbol{x} ) | 12 |
| 1621 | ( int_{0}^{1} frac{x d x}{left(x^{2}+1right)^{2}}= ) A ( cdot 1 / 2 ) в. ( 1 / 3 ) c. ( 1 / 4 ) D. | 12 |
| 1622 | Evaluate: ( int_{2}^{3} frac{x d x}{x^{2}+1} ) | 12 |
| 1623 | ( int_{0}^{pi / 2} frac{d x}{3+4 sin x} d x= ) ( frac{1}{sqrt{(k)}} log left(frac{4+sqrt{7}}{3}right) ) then find the value of ( k ) | 12 |
| 1624 | If ( I=int_{0}^{pi} frac{d x}{5+3 cos x} ) then ( I ) equals ( A ) в. ( 2 pi / 3 ) c. ( pi / 4 ) D. 2pi/sqrt sqrt | 12 |
| 1625 | Evaluate the integral ( int_{-1}^{1} frac{d x}{x^{2}+2 x+5} ) using substitution | 12 |
| 1626 | Using Mean-Value Theorem, the best upper bound of ( int_{0}^{1} frac{sin x}{1+x^{2}} d x ) is A ( cdot frac{pi}{4} sin 1 ) B. ( pi sin 1 ) c. ( frac{pi}{2} sin 1 ) D. ( frac{pi}{4} sin left(frac{1}{2}right) ) | 12 |
| 1627 | Consider the integrals ( A=int_{0}^{pi} frac{sin x d x}{sin x+cos x} ) and ( B= ) ( int_{0}^{pi} frac{sin x d x}{sin x-cos x} ) Which one of the following is correct? A. ( A=2 B ) в. ( B=2 A ) c. ( A=B ) D. ( A=3 B ) | 12 |
| 1628 | Find the integral ( int frac{2 x}{xleft(x^{2}+1right)} d x ) | 12 |
| 1629 | If ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{f}(boldsymbol{x})=4 boldsymbol{x}^{3}-frac{boldsymbol{3}}{boldsymbol{x}^{4}} ) such that ( boldsymbol{f}(boldsymbol{2})= ) 0. Then ( f(x) ) is | 12 |
| 1630 | Find area of ( boldsymbol{y}=boldsymbol{x}^{2} ) from ( boldsymbol{x}=boldsymbol{2} ) to ( boldsymbol{x}=boldsymbol{4} ) | 12 |
| 1631 | Evaluate ( int frac{x cos ^{-1} x}{sqrt{1-x^{2}}} d x ) | 12 |
| 1632 | Evaluate ( int frac{cos x+x sin x}{x^{2}+cos ^{2} x} d x ) A ( cdot-tan ^{-1}left(frac{cos x}{x}right)+c ) в. ( ln left(frac{x+cos x}{x}right)+c ) c. ( tan ^{-1}left(frac{x}{x+cos x}right)+c ) D. None | 12 |
| 1633 | ( int_{0}^{pi / 4} log (1+tan x) d x ) is equal to A ( cdot frac{pi}{8} log _{e} 2 ) в. ( frac{pi}{4} log _{e} e ) c. ( frac{pi}{4} log _{e} 2 ) D ( cdot frac{pi}{8} log _{e}left(frac{1}{2}right) ) | 12 |
| 1634 | ( int x^{x} ln (x) d x= ) A ( cdot frac{x^{x}}{ln x}+c ) B . ( x^{x}+c ) c. ( frac{x^{x+1}}{x+1}+c ) D. none of these | 12 |
| 1635 | Evaluate ( : intleft(x+frac{1}{x}right)^{3} d x, x>0 ) | 12 |
| 1636 | Solve ( : int_{0}^{frac{pi}{2}} tan ^{5} x cos ^{8} x d x ) | 12 |
| 1637 | Evaluate ( int frac{d x}{sqrt{5 x^{2}-2 x}} ) | 12 |
| 1638 | Which of the following is true for indefinite integral? This question has multiple correct options | 12 |
| 1639 | The angle made by the tangent line at (1 3) on the curve ( y=4 x-x^{2} ) with ( overline{O X} ) is ( A cdot tan ^{-1} 2 ) B. ( tan ^{-1}(1 / 2) ) ( c cdot tan ^{-1}-2 ) D. None of these | 12 |
| 1640 | Evaluate the following integral ( int frac{1}{x(3+log x)} d x ) | 12 |
| 1641 | ( frac{boldsymbol{x}+mathbf{1}}{(mathbf{2} boldsymbol{x}-mathbf{1})(mathbf{3} boldsymbol{x}+mathbf{1})}=frac{boldsymbol{A}}{mathbf{2} boldsymbol{x}-mathbf{1}}+ ) ( frac{B}{3 x+1} Rightarrow 16 A+9 B= ) ( A ) B. 5 ( c cdot 6 ) ( D ) | 12 |
| 1642 | Evaluate: ( int frac{sin ^{2} x}{cos ^{4} x} d x ) A ( cdot frac{1}{2} tan ^{2} x+c ) B ( cdot frac{1}{2} cot ^{2} x+c ) c. ( frac{1}{3} cot ^{3} x+c ) D. ( frac{1}{3} tan ^{3} x+c ) | 12 |
| 1643 | ( int_{a}^{b} cos x d x ) Obtain the definite integral as a limit of d sum. | 12 |
| 1644 | Evaluate: ( int sqrt{frac{boldsymbol{x}-mathbf{5}}{boldsymbol{x}-mathbf{9}}} boldsymbol{d} boldsymbol{x} ) | 12 |
| 1645 | Solve ( int frac{cos ^{2} theta d theta}{cos ^{2} theta+4 sin ^{2} theta} ) | 12 |
| 1646 | 19. Let T>Obe a fixed real number. Suppose fis a continuous function such that for all X ER,f(x+T)=f(x). IF I=1 $5)dz then the value of f(2x)dx is (2002) (2) 3/21 (6) 21 (1) 31 (2) 1 3 | 12 |
| 1647 | The value of ( int frac{d x}{sin x cdot sin (x+alpha)} ) is equal to | 12 |
| 1648 | The value of ( int_{0}^{pi / 2} frac{sin 2 t}{sin ^{4} t+cos ^{4} t} d t ) ( A ) в. c. D. ( frac{pi}{2} ) | 12 |
| 1649 | 40. The value of x? COS X dx is equ -dx is equal to (JEE Adv. 2016) 1+e* (b) * +2 -2 (C) 1 -e ž T² + e2 | 12 |
| 1650 | ( int sec x ln (sec x+tan x) d x ) | 12 |
| 1651 | ( int x cos ^{-1} x d x ) | 12 |
| 1652 | ( int_{2-ell n 3}^{3+ell n 3} frac{ln (4+x)}{ell n(4+x)+ell n(9-x)} d x ) is equal to: A. cannot be evaluated B. is equal to ( frac{5}{2} ) c. is equal to ( 1+2 ell n 3 ) D. is equal to ( frac{1}{2}+ell n ) 3 | 12 |
| 1653 | Evaluate the given integral. ( int e^{x}left(frac{sin x cos x-1}{sin ^{2} x}right) d x ) | 12 |
| 1654 | If differential equation of family of curves ( boldsymbol{y} ln |boldsymbol{c} boldsymbol{x}|=boldsymbol{x}, ) where ( c ) is an arbitrary constant, is ( boldsymbol{y}^{prime}=frac{boldsymbol{y}}{boldsymbol{x}}+boldsymbol{phi}left(frac{boldsymbol{x}}{boldsymbol{y}}right) ) for some function ( phi ), then ( phi(2) ) is equal to? | 12 |
| 1655 | I et f(x)=7tanⓇx+7tan x-3tan4x – 3tanPx for all x el Then the correct expression(s) is(are) (JEE Adv. 2015). T/4 – T/4 xf (x) dx = f(x) dx = 0 0 (2) S (x) dx = 1 / 2 (0) 1 + (x) dx = 1 / 6 i redde = 0 (0 s sodas = 1 T/4 T/4 | 12 |
| 1656 | ( int_{0}^{1} frac{x e^{x}}{(1+x)^{2}} d x ) | 12 |
| 1657 | Evaluate the following as the limit of sum : ( int_{0}^{2}(x+4) d x ) A .4 B. 6 c. 8 D. 10 | 12 |
| 1658 | Evaluate : ( int frac{left(x^{4}-xright)^{1 / 4}}{x^{5}} d x ) | 12 |
| 1659 | ( int e^{x} 2^{3 log _{2} x} d x=e^{x} f(x)+c, ) then ( boldsymbol{f}(boldsymbol{x})= ) A. ( x^{3}-3 x^{2}+6 x-6 ) B. ( x^{3}-3 x^{2}-6 x-3 ) c. ( x^{3}-3 x^{2}+6 x+6 ) D. ( x^{3}+3 x^{2}+6 x+6 ) | 12 |
| 1660 | Find ( int sqrt{boldsymbol{x}}left(boldsymbol{3} boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{3}right) boldsymbol{d} boldsymbol{x} ) | 12 |
| 1661 | 16. Evaluate (log[V1- x + V1+x]dx | 12 |
| 1662 | ( int frac{1}{sin ^{2} x+sin 2 x} d x ) A. ( frac{1}{2} log frac{tan x}{tan x+2} ) B. ( -frac{1}{2} log frac{tan x}{tan x+2} ) c. ( frac{1}{2} log frac{tan x}{tan x-2} ) D. ( log frac{tan x}{tan x+2} ) | 12 |
| 1663 | Assertion If ( n>1 ) then ( int_{0}^{infty} frac{d x}{1+x^{n}}= ) ( int_{0}^{1} frac{d x}{left(1-x^{n}right)^{1 / n}} ) Reason ( int_{a}^{b} f(x) d x=int_{a}^{b} f(a+b-x) d x ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1664 | Evaluate ( int_{1}^{4}(x-1) d x ) | 12 |
| 1665 | ntegrate the function ( frac{sec ^{2} x}{sqrt{tan ^{2} x+4}} ) | 12 |
| 1666 | Evaluate the following integral ( int frac{sec x operatorname{cosec} x}{log (tan x)} d x ) | 12 |
| 1667 | Evaluate the integral: ( int_{0}^{2} frac{x}{3} d x ) | 12 |
| 1668 | Solve the integral ( int sqrt{frac{1+x}{1-x}} d x ) | 12 |
| 1669 | ( int_{0}^{pi} frac{d x}{1+2^{tan x}}= ) ( A cdot O ) B . ( pi / 4 ) c. ( pi / 2 ) D. | 12 |
| 1670 | Evaluate the following definite integral ( int_{0}^{pi / 2} frac{cos ^{2} x}{1+3 sin ^{2} x} d x ) | 12 |
| 1671 | ( int frac{sec ^{2} x d x}{sqrt{tan ^{2} x+4}}= ) A ( cdot ln (tan x+sqrt{tan ^{2} x+4})+C ) B. ( frac{1}{2} ln (tan x+sqrt{tan ^{2} x+4})+C ) c. ( ln left(frac{1}{2} tan x+frac{1}{2} sqrt{tan ^{2} x+4}right)+C ) D. None of these | 12 |
| 1672 | The value of ( int_{0}^{frac{pi}{2}} frac{2^{sin x}}{2^{sin x}+2^{cos x}} d x ) is A. B. ( c cdot 0 ) D. none of these | 12 |
| 1673 | ( int_{0}^{frac{pi}{4}} sin 3 x sin 2 x d x ) | 12 |
| 1674 | Evaluate ( int frac{d x}{x^{2}-4 x+13} ) | 12 |
| 1675 | ( int cos ^{2} theta d theta=? ) | 12 |
| 1676 | Evaluate the integrals: ( int(2 x+9)^{5} d x ) | 12 |
| 1677 | ( int_{-1}^{1} frac{d x}{x^{2}+2 x+5} ) | 12 |
| 1678 | ( int frac{1}{x+x log x} d x ) | 12 |
| 1679 | Evaluate the following integral ( int frac{sec x operatorname{cosec} x}{log (tan x)} d x ) | 12 |
| 1680 | ( intleft(1+x-x^{-1}right) e^{x+x^{-1}} d x ) is equal to A ( cdot(x+1) e^{x+x^{-1}}+C ) B . ( (x-1) e^{x+x^{-1}}+C ) c. ( x e^{x+x^{-1}}+C ) D. ( x e^{x+x^{-1}} x+C ) | 12 |
| 1681 | ( int e^{2 x}left(frac{1+sin 2 x}{1+cos 2 x}right) d x ) is equal to A ( cdot e^{2 x} tan x+C ) B . ( e^{2 x} cot x+C ) ( ^{mathrm{c}} cdot frac{e^{2 x} tan x}{2}+C ) ( ^{mathrm{D} cdot frac{e^{2 x} cot x}{2}}+C ) | 12 |
| 1682 | ( int_{0}^{1} frac{d x}{x sqrt{x}} ) ( mathbf{A} cdot 2 ) B . -2 ( mathbf{c} cdot 1 ) D. 3 | 12 |
| 1683 | ( boldsymbol{I}=int frac{x+2}{(x+1)^{2}} boldsymbol{d} boldsymbol{x} ; ) then I is equal to ( mathbf{A} cdot log (x+1)+frac{1}{x+1}+c ) B. ( log (x+2)-frac{1}{x+1}+c ) c. ( log (1+x)-frac{1}{x+1}+c ) D. ( log (x+2)+frac{1}{x+1}+c ) | 12 |
| 1684 | Show that ( int_{0}^{1} frac{log x}{sqrt{left(1-x^{2}right)}} d x=frac{pi}{2} cdot log frac{1}{2} ) | 12 |
| 1685 | The value of integral ( int_{-1}^{3}left(tan ^{-1}left(frac{x}{1+x^{2}}right)+tan ^{-1}left(frac{x^{2}+1}{x}right)right. ) | 12 |
| 1686 | 34. Let p(x) be a function defined on R such that p'(x) =p'(1-x), for all x e [0, 1],p (O)= 1 and p (1)=41. Then p(x) dx equals [2010] (a) 21 (6) 41 (c) 42 (d) VAI | 12 |
| 1687 | Evaluate ( int frac{x+2}{sqrt{x^{2}+4 x+1}} cdot d x ) | 12 |
| 1688 | Evaluate ( int frac{d x}{left(x^{2}+1right) sqrt{x^{2}+1}} ) | 12 |
| 1689 | If ( boldsymbol{I}=int frac{boldsymbol{x}^{2}+boldsymbol{a}^{2}}{boldsymbol{x}^{4}-boldsymbol{a}^{2} boldsymbol{x}^{2}+boldsymbol{a}^{4}} boldsymbol{d} boldsymbol{x} ) A ( cdot frac{1}{a} tan ^{-1}left(frac{a x}{x^{2}-a^{2}}right)+C ) B. ( frac{1}{a} tan ^{-1}left(frac{x^{2}-a^{2}}{a x}right)+C ) c. ( log |x+sqrt{x^{2}-a^{2}}|+x+C ) D. none of these | 12 |
| 1690 | Evaluate the integral ( int_{-1}^{1}(sqrt{1-x+x^{2}}-sqrt{1+x+x^{2}}) d x ) ( A cdot frac{1}{2} ) в. c. 0 ( D ) | 12 |
| 1691 | V2 ve 2) The option(s) with the values of a and L that satisfy the following equation is(are) (JEE Adv. 2015) 47 et (sinºat + cos4 at dt -=L? fet (sinºat + cos^ at )dt 41 – 1 (a) a=2, L= et – 1 4r +1 (b) a=2, L= et +1 e4T +1 (d) a=4,L=; et +1 ©) a=4,L=*1 (a) a=4,L= ** *1 (c) at, ber – 1 | 12 |
| 1692 | 20.5(_3 +2x )de is equal to.. | 12 |
| 1693 | Evaluate ( int frac{2 cos x-3 sin x}{4 cos x+5 sin x} d x ) | 12 |
| 1694 | Integrate: ( int sin x^{2} d x ) | 12 |
| 1695 | Evaluate : ( int_{-1}^{1} frac{1}{x^{2}+2 x+5} d x ) | 12 |
| 1696 | ( int frac{x+3}{(x-1)(x-2)(x-3)} d x ) | 12 |
| 1697 | If ( f^{prime}(x)=x+frac{1}{x}, ) then value of ( f(x) ) is A ( cdot x^{2}+log x+c ) в. ( frac{x^{2}}{2}+log x+c ) c. ( frac{x}{2}+log x+c ) D. None of these | 12 |
| 1698 | Evaluate the following integrals: ( int cot ^{5} x operatorname{cosec}^{4} x d x ) | 12 |
| 1699 | ( int frac{x d x}{(x-1)(x-2)} ) equals ( ^{mathbf{A}} cdot log left|frac{(x-1)^{2}}{x-2}right|+C ) ( ^{mathbf{B}} cdot log left|frac{(x-2)^{2}}{x-1}right|+C ) ( ^{mathrm{c}} log left(left(frac{x-1}{x-2}right)^{2} mid+Cright. ) D. ( log |(x-1)(x-2)|+C ) | 12 |
| 1700 | Evaluate ( int_{4}^{12} x d x ) | 12 |
| 1701 | ( f frac{1}{xleft(x^{2}+a^{2}right)}=frac{A}{x}+frac{B x+C}{x^{2}+a^{2}}, ) then ( tan ^{-1}left(frac{A}{B}right)= ) A ( cdot frac{3 pi}{4} ) в. ( c cdot-frac{pi}{4} ) D. | 12 |
| 1702 | The value of ( int frac{x^{2}+1}{x^{4}-x^{2}+1} d x ) is A ( cdot tan ^{-1}left(2 x^{2}-1right)+C ) B. ( tan ^{-1}left(frac{x^{2}-1}{x}right)+C ) ( ^{mathrm{c}} cdot sin ^{-1}left(x-frac{1}{x}right)+C ) D. ( tan ^{-1} x^{2}+C ) | 12 |
| 1703 | The value of the intgral ( int_{-pi / 2}^{pi / 2} log left(frac{a-sin theta}{a+sin theta}right) d theta, a>0 ) is? ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 5 | 12 |
| 1704 | ( int_{0}^{1} x e^{x} d x= ) ( A cdot 1 ) B. 2 ( c .3 ) D. | 12 |
| 1705 | Solve: ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x ) | 12 |
| 1706 | ( boldsymbol{I}=int sqrt{frac{a+x}{a-x}} d x ) | 12 |
| 1707 | ( sqrt[*]{frac{x}{1+x^{4}}} d x ) ( mathbf{A} cdot tan ^{-1} x^{2} ) B. ( 2 tan ^{-1} x^{2} ) c. ( frac{1}{2} tan ^{-1} x^{2} ) D. ( frac{1}{2} tan ^{-1} x ) | 12 |
| 1708 | ( operatorname{Let} boldsymbol{I}_{n}=int_{0}^{pi} frac{sin ^{2}(boldsymbol{n} boldsymbol{x})}{sin ^{2} boldsymbol{x}} boldsymbol{d} boldsymbol{x}, boldsymbol{n} in boldsymbol{N}, ) then A ( cdot I_{n+2}+I_{n}=21_{n+1} ) В. ( I_{n}=I_{n+1} ) ( mathbf{c} cdot I_{n}=n pi ) D. ( I_{1}, I_{I}, I_{3}, ldots ldots I_{n} ) are in harmonic progression | 12 |
| 1709 | 41. The integral f/1+4 sin? * – 4sin dx equals: [JEEM 201 (a) 413-4. (b) 483-4– (2) 27 – 4 – 4√3 27 (c) r-4 | 12 |
| 1710 | Find the area of the figure bounded by the following curves Find all values of a for which the inequality ( int_{0}^{a} x d x leqslant a+4 ) is satisfied. | 12 |
| 1711 | Integrate: ( int_{0}^{pi} frac{d x}{5+3 cos x} ) | 12 |
| 1712 | If ( f(x) ) and ( g(x) ) be continuous functions over the closed interval ( [mathbf{0}, boldsymbol{a}] ) such that ( boldsymbol{f}(boldsymbol{x})=boldsymbol{f}(boldsymbol{a}-boldsymbol{x}) ) and ( boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{a}-boldsymbol{x})=mathbf{2} . ) Then ( int_{0}^{a} f(x) dot{g}(x) d x ) is equal to A ( cdot int_{0}^{a} f(x) d x ) в. ( int_{0}^{a} g(x) d x ) ( c cdot 2 a ) D. none of these | 12 |
| 1713 | Let ( boldsymbol{f}(boldsymbol{x})=frac{1}{3} cot ^{3} boldsymbol{x}-cot boldsymbol{x}+ ) ( int cot ^{4} x d x ) and ( fleft(frac{pi}{2}right)=frac{pi}{2}, ) then ( boldsymbol{f}(boldsymbol{x})= ) ( mathbf{A} cdot pi-x ) B. ( x-pi ) c. ( frac{pi}{2}-x ) D. ( x ) | 12 |
| 1714 | Integrate ( int_{a}^{b} cos x d x ) | 12 |
| 1715 | ( int_{0}^{pi / 2} frac{d x}{1+tan x} ) This question has multiple correct options A . a multiple of ( pi / 4 ) B. a multiple of ( pi / 2 ) c. equal to ( pi / 4 ) D. a multiple of ( pi ) | 12 |
| 1716 | ( int_{1}^{2}left(frac{x-1}{x^{2}}right) e^{x} d x ) | 12 |
| 1717 | If ( frac{(1+x)(1+2 x)(1+3 x)}{(1-x)(1-2 x)(1-3 x)}=K+ ) ( frac{mathbf{A}}{mathbf{1}-mathbf{x}}+frac{mathbf{B}}{mathbf{1}-mathbf{2} mathbf{x}}+frac{mathbf{C}}{mathbf{1}-mathbf{3} mathbf{x}}, ) then which of the following is correct ( mathbf{A} cdot mathbf{K}=6 ) B. ( A=12 ) c. ( mathrm{B}=30 ) ( mathbf{D} cdot mathbf{C}=-20 ) | 12 |
| 1718 | The value of the integral ( int_{-pi}^{pi}(cos a x- ) ( sin b x)^{2} d x, ) where ( a ) and ( b ) are integers, is A ( .2 pi(1+a+b) ) B. 0 c. ( pi ) D. ( 2 pi ) | 12 |
| 1719 | Evaluate the following integral ( int frac{operatorname{cosec} x}{log tan frac{x}{2}} d x ) | 12 |
| 1720 | Integrate ( int frac{d x}{(x+1)(x+5)} ) | 12 |
| 1721 | The value of ( int frac{log x}{(x+1)^{2}} d x ) is A ( cdot frac{-log x}{x+1}+log x-log (x+1)+C ) B. ( frac{log x}{x+1}+log x-log (x+1)+C ) c. ( frac{log x}{x+1}-log x-log (x+1)+C ) D. ( frac{-log x}{x+1}-log x-log (x+1)+C ) | 12 |
| 1722 | If the primitive of ( frac{e^{x}left(1+e^{x}right)}{sqrt{1-e^{2 x}}} ) is ( boldsymbol{f} boldsymbol{o} boldsymbol{g}(boldsymbol{x})-sqrt{boldsymbol{h}(boldsymbol{x})}+boldsymbol{C} ) then This question has multiple correct options A ( cdot f(x)=sin ^{-1} x ) B ( cdot g(x)=e^{2 x} ) ( mathbf{c} cdot g(x)=e^{x} ) D. ( h(x)=1-e^{2 x} ) | 12 |
| 1723 | ( int sqrt{1+x^{2}} d x ) is equal to A ( cdot frac{x}{2} sqrt{1+x^{2}}+frac{1}{2} log |+sqrt{1+x^{2}}|+C ) B. ( frac{2}{3}left(1+x^{2}right)^{frac{2}{3}}+C ) c. ( frac{2}{3} xleft(1+x^{2}right)^{frac{3}{2}}+C ) D ( cdot frac{x^{2}}{2} sqrt{1+x^{2}}+frac{1}{2} x^{2} log |x+sqrt{1+x^{2}}|+C ) | 12 |
| 1724 | Evaluate ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x ) | 12 |
| 1725 | Integrate: ( int frac{x}{x^{4}-x^{2}+1} d x ) | 12 |
| 1726 | If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}}+int_{0}^{1}left(boldsymbol{e}^{boldsymbol{x}}+boldsymbol{t} boldsymbol{e}^{-boldsymbol{x}}right) boldsymbol{f}(boldsymbol{t}) boldsymbol{d} boldsymbol{t} ) then prove that ( f(x)=frac{2(e-1)}{4 e-2 e^{2}} cdot e^{x}+ ) ( frac{e-1}{4-2 e} cdot e^{-x} ) | 12 |
| 1727 | Evaluate ( int x e^{x} d x ) A ( .-x e^{x}-e^{x}+c ) B. ( -x e^{x}+e^{x}+c ) c. ( x e^{x}+e^{x}+c ) D. ( x e^{x}-e^{x}+c ) | 12 |
| 1728 | Evaluate ( intleft(frac{x^{6}-1}{1+x^{2}}right) d x ) for ( x in R ) | 12 |
| 1729 | Solve ( int frac{2 x ln left(x^{2}-1right)}{left(x^{2}-1right)} d x ) | 12 |
| 1730 | The value of ( int_{0}^{infty} frac{d x}{left(x^{2}+4right)left(x^{2}+9right)} ) is A ( cdot frac{pi}{60} ) в. ( frac{pi}{20} ) c. ( frac{pi}{40} ) D. ( frac{pi}{80} ) | 12 |
| 1731 | ( int frac{1-x^{7}}{xleft(1+x^{7}right)} d x ) equals A ( quad ln |x|+frac{2}{7} ln left|1+x^{7}right|+c ) B ( cdot ln |x|+frac{2}{4} ln left|1-x^{7}right|+c ) C ( quad ln |x|-frac{2}{7} ln left|1+x^{7}right|+c ) D. ( -ln |x|+frac{2}{4} ln left|1-x^{7}right|+c ) | 12 |
| 1732 | ( int_{100}^{2014} frac{sqrt{x}}{sqrt{2114-x}+sqrt{x}} d x= ) A . 1914 в. 957 ( c .1007 ) D. ( frac{2015}{2} ) | 12 |
| 1733 | Evaluate the following definite integral: ( int_{0}^{pi / 2} sin x cos x d x ) | 12 |
| 1734 | Evaluate ( int_{0}^{1} x^{4} d x ) | 12 |
| 1735 | Prove that ( int frac{x^{2}}{x^{6}+1} d x ) | 12 |
| 1736 | Assertion If ( boldsymbol{f}, boldsymbol{g} ) and ( boldsymbol{h} ) be continuous function on ( [0, a] ) such that ( f(x)=f(a-x) ) ( g(x)=-g(a-x) ) and ( 3 h(x)-4 h(a- ) ( boldsymbol{x})=mathbf{5}, ) then ( int_{0}^{boldsymbol{a}} boldsymbol{f}(boldsymbol{x}) boldsymbol{g}(boldsymbol{x}) boldsymbol{h}(boldsymbol{x}) boldsymbol{d} boldsymbol{x}=mathbf{0} ) Reason ( int_{0}^{a} f(x) g(x) d x=0 ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1737 | ( int_{0}^{pi / 2} frac{sin ^{6} x}{cos ^{6} x+sin ^{6} x} d x ) is equal to A . 0 в. ( pi ) c. ( frac{pi}{4} ) D . ( 2 pi ) | 12 |
| 1738 | If ( f(x)=int frac{x^{2}+sin ^{2} x}{1+x^{2}} cdot sec ^{2} x d x ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} ) then ( boldsymbol{f}(mathbf{1})= ) A ( cdot 1-frac{pi}{4} ) B. ( frac{pi}{4}-1 ) c. ( tan 1-frac{pi}{4} ) D. None of these | 12 |
| 1739 | Evaluate ( : int x tan ^{-1} x d x ) | 12 |
| 1740 | Evaluate: ( int_{0}^{pi / 2} cos ^{2} x d x ) | 12 |
| 1741 | Evaluate the following definite integral. ( int_{2}^{3} frac{x}{x^{2}+1} d x ) | 12 |
| 1742 | ( int frac{x^{3}-x^{2}+x-1}{x-1} d x ) | 12 |
| 1743 | If ( y=2^{2} 3^{2 x} 5^{-5} 7^{-5} ) then ( frac{d y}{d x}= ) | 12 |
| 1744 | Integrate the function ( x sin 3 x ) | 12 |
| 1745 | If ( boldsymbol{I}_{n}=int_{0}^{sqrt{3}} frac{boldsymbol{d} boldsymbol{x}}{1+boldsymbol{x}^{n}},(boldsymbol{n}=mathbf{1}, mathbf{2}, mathbf{3}, boldsymbol{4} dots dots dots) ) then find the value of ( lim _{n rightarrow infty} I_{n}, ) is A . B. 1 c. 2 D. ( frac{1}{2} ) | 12 |
| 1746 | ( int frac{[log (log boldsymbol{x})]^{boldsymbol{m}}}{boldsymbol{x} log boldsymbol{x}} boldsymbol{d} boldsymbol{x}= ) ( mathbf{A} cdot frac{[log (log (x))]^{m+1}}{m+1}+c ) B. ( frac{[log (log (x))]^{m}}{m}+c ) c. ( frac{[log (log (x))]^{m+1}}{m}+c ) D. ( frac{[log (log (x))]^{m}}{m+1}+c ) | 12 |
| 1747 | Solve ( int x^{3} log x d x ) A ( frac{x^{4} log x}{4}+C ) B. ( I=frac{x^{4}}{4} log x-frac{x^{4}}{16}+C ) c. ( frac{1}{8}left[x^{4} log x-4 x^{2}right]+C ) D ( cdot frac{1}{16}left[4 x^{4} log x+x^{4}right]+C ) | 12 |
| 1748 | f (x+1) 11. Evaluate x01 + xet 2 dx. | 12 |
| 1749 | 18. For, a eR, al > 1, let lim =54 1+√2+….. + In 7/31 1 1 (an +1)2 (an + 2)2 1 1)’ (antov +…+ (an+n)? )) (JEE Adv. 2019) Then the possible value(s) of a is/are (a) – 9 (b) 7 (c) – 6 (d) 8 | 12 |
| 1750 | Prove that ( int_{0}^{frac{32 pi}{3}} sqrt{1+cos 2 x} d x= ) ( 22 sqrt{2}-sqrt{frac{3}{2}} ) | 12 |
| 1751 | ( int frac{d x}{x(1+sqrt[3]{x})^{2}} ) is equal to A ( cdotleft(log frac{x^{1 / 3}}{1+x^{1 / 3}}+frac{1}{1+sqrt[3]{x}}right)+c ) B. ( 3left(log frac{1+sqrt[3]{x}}{sqrt[3]{x}}+frac{1}{1+sqrt[3]{x}}right)+c ) ( ^{mathrm{C}} cdotleft(log frac{x^{1 / 3}}{1+x^{1 / 3}}-frac{1}{1+sqrt[3]{x}}right)+c ) D. ( 3left(log frac{1+sqrt[3]{x}}{sqrt[3]{x}}-frac{1}{1+sqrt[3]{x}}right)+c ) | 12 |
| 1752 | Integrate ( int frac{x^{2} d x}{x^{6}-a^{6}} d x ) | 12 |
| 1753 | ( frac{x^{2}+x+1}{(x-1)(x-2)(x-3)}=frac{A}{x-1}+ ) ( frac{B}{x-2}+frac{C}{x-3} ) ( Rightarrow boldsymbol{A}+boldsymbol{C}= ) ( A cdot 4 ) B. 5 ( c cdot 6 ) D. | 12 |
| 1754 | ( int e^{2 x-3}+7^{4-3(x / 2)}+sin left(3 x-frac{1}{2}right)+ ) ( cos left(frac{2}{5} x-2right)+a^{3 x+2} d x ) | 12 |
| 1755 | Evaluate : ( int frac{(3 x+5) d x}{sqrt{x^{2}+4 x+3}} ) | 12 |
| 1756 | Evaluate ( int frac{d x}{9 x^{2}+6 x+5} ) | 12 |
| 1757 | Integrate: ( int frac{x^{4}}{x^{2}+1} d x= ) A ( cdot frac{x^{3}}{3}-x+tan ^{-1} x+c ) B. ( frac{x^{5}}{5}+tan ^{-1} x+c ) C ( cdot 4 x^{3}+tan ^{-1} x+c ) D ( frac{x^{4}}{4}-x+tan ^{-1} x+c ) | 12 |
| 1758 | 32. [cot x]dx , where [.] denotes the greatest integer function, is equal to : [2009] (a) 1 (1) 1 (c) – – – | 12 |
| 1759 | [(x+)3 + cos(x +31)]dx is equal to (b) 323 (d) -1 | 12 |
| 1760 | ( int sqrt{frac{1-x}{1+x}} d x= ) | 12 |
| 1761 | ( int frac{x}{left(x^{2}-a^{2}right)left(x^{2}-b^{2}right)} d x ) is equal to ( frac{1}{kleft(b^{2}-a^{2}right)} ln frac{left(x^{2}-b^{2}right)}{left(x^{2}-a^{2}right)} cdot ) Find ( k ) | 12 |
| 1762 | Evaluate: ( int frac{2 x}{x^{2}} d x ) | 12 |
| 1763 | ( int sqrt{1+sin x} d x= ) A ( cdot frac{1}{2}left(sin frac{x}{2}+cos frac{x}{2}right)+c ) B ( cdot frac{1}{2}left(sin frac{x}{2}-cos frac{x}{2}right)+c ) c. ( 2 sqrt{1+sin x}+c ) D. ( -2 sqrt{1-sin x}+c ) | 12 |
| 1764 | Solve ( intleft(frac{x-1}{x+1}right)^{4} d x= ) | 12 |
| 1765 | If ( I=int frac{x^{2}}{(x-a)(x-b)} d x, ) then equals A ( cdot x+frac{1}{a-b} log left|frac{x-a}{x-b}right|+C ) в. ( quad x+frac{1}{a-b} log left|frac{x-a}{x-b}right|^{a^{2}+b^{2}}+C ) c. ( x+frac{1}{a-b}left{a^{2} log |x-a|-b^{2} log |x-b|right}+C ) D. none of these | 12 |
| 1766 | Evaluate the following definite integral: ( int_{0}^{1} x+x^{2} d x ) | 12 |
| 1767 | For the function ( f(x)=e^{x}, a=0, b=1 ) the value of ( c ) in mean value theorem will be ( mathbf{A} cdot mathbf{0} ) B. ( log (e-1) ) ( c cdot log x ) ( D ) | 12 |
| 1768 | Evaluate ( int frac{sin theta}{sin 3 theta} d theta ) | 12 |
| 1769 | Evaluate: ( int_{0}^{2}left[x^{2}right] d x ) | 12 |
| 1770 | Evaluate: ( int frac{x^{4}+4}{x^{2}-2 x+2} d x ) A ( cdot frac{x^{3}}{2}+x^{2}+2 x+C ) B. ( frac{x^{3}}{3}+x^{2}+2 x+C ) c. ( frac{x^{3}}{3}+x^{2}+x+C ) D. ( frac{x^{3}}{3}+x^{2}-2 x+C ) | 12 |
| 1771 | Solve ( int_{0}^{2}left(x^{2}+1right) d x ) A ( cdot frac{2}{3} ) в. ( frac{14}{3} ) ( c cdot frac{8}{3} ) D. ( frac{11}{3} ) | 12 |
| 1772 | ( int frac{left(x^{3}+8right)(x-1)}{x^{2}-2 x+4} d x ) | 12 |
| 1773 | Evaluate: ( int_{-pi / 3}^{pi / 3} cos ^{2} x d x ) A. ( sqrt{3} / 4 ) в. ( pi / 3 ) ( ^{C} cdot frac{pi}{3}+frac{sqrt{3}}{4} ) D. ( frac{pi}{3}-frac{sqrt{3}}{4} ) | 12 |
| 1774 | TO xf (sin x)dx is equal to [2006] cu af scos din os x) dx af sains f(sin x)dx /2 I f(sin x)dx. (d) a j f (cos x)dx | 12 |
| 1775 | Integrate the function ( frac{x+2}{sqrt{4 x-x^{2}}} ) | 12 |
| 1776 | Find the value of ( int_{0}^{frac{pi}{2}} log (tan x) d x ) | 12 |
| 1777 | Solve ( int_{b}^{a} frac{x}{sqrt{a^{2}+x^{2}}} d x ) | 12 |
| 1778 | 1. If S* (t)dt=x+S + f(t) dt, then the value of f(1) is (1998 – 2 Man | 12 |
| 1779 | Evaluate: ( int_{0}^{pi / 4} log (1+tan theta) d theta ) | 12 |
| 1780 | Evaluate ( int_{0}^{pi / 2} cos x d x ) | 12 |
| 1781 | Evaluate: ( intleft(3 x^{2}-5right)^{2} d x ) | 12 |
| 1782 | 14. (cos – sinu) a = (b) 2 cos25+ c (c) (cos 13)+c () x-cos x + c (a) x + cos x + c COS (a ) x-COS X + C | 12 |
| 1783 | ( int frac{1}{(2 x+1) sqrt{x^{2}-x-2}} d x= ) A. ( -frac{1}{sqrt{5}} sin ^{-1} frac{7+4 x}{3(2 x+1)}+c ) B. ( -frac{1}{sqrt{5}} cos frac{7+4 x}{3(2 x+1)}+c ) c. ( -frac{1}{sqrt{5}} sinh ^{-1} frac{7+4 x}{3(2 x+1)}+c ) D. ( -frac{1}{sqrt{5}} cosh ^{-1} frac{7+4 x}{3(2 x+1)}+c ) | 12 |
| 1784 | Solve: ( int_{0}^{1} frac{1}{2 x^{2}+x+1} d x ) | 12 |
| 1785 | solve : ( int(a x+b)^{2} d x ) | 12 |
| 1786 | Evaluate i) ( int frac{x^{2}+1}{x^{4}+1} d x ) ( i i) int frac{d x}{x^{2}+1} ) | 12 |
| 1787 | ( int sin x cdot cos x d x ) | 12 |
| 1788 | The value of the integral ( int_{-2}^{2}(1+ ) ( 2 sin x) e^{|x|} d x ) is equal to A . B. ( e^{2}-1 ) c. ( 2left(e^{2}-1right) ) D. | 12 |
| 1789 | Evaluate: ( int e^{x}(tan x+log (sec x)) d x ) | 12 |
| 1790 | Integrate: ( frac{e^{2 x}-1}{e^{2 x}+1} ) | 12 |
| 1791 | Integrate the function ( sqrt{x^{2}+4 x-5} ) | 12 |
| 1792 | Integrate ( : int frac{1+tan x}{x+log sec x} d x ) | 12 |
| 1793 | Find the following integrals: ( intleft(sqrt{boldsymbol{x}}-frac{1}{sqrt{x}}right)^{2} d x ) | 12 |
| 1794 | ( int cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right) d x ) | 12 |
| 1795 | ( int_{0}^{pi / 2} sin x cos x d x ) is equal to: | 12 |
| 1796 | Evaluate the given integral. ( int frac{1}{xleft(x^{3}+8right)} d x ) | 12 |
| 1797 | ( boldsymbol{I} boldsymbol{f} boldsymbol{I}=int boldsymbol{x} sqrt{frac{x^{2}+1}{x^{2}-1}} boldsymbol{d} boldsymbol{x}, ) then I equals A ( cdot frac{1}{2} sqrt{x^{4}-1}+frac{1}{2} sqrt{x^{4}+1}+c ) B – ( frac{1}{2} sqrt{x^{4}-1}+frac{1}{2} operatorname{tn}left(x^{2}+sqrt{x^{4}-1}right)+c ) c. ( sqrt{x^{4}-1}+sin ^{-1}left(x^{2}right)+c ) D. ( sqrt{x^{4}-1}+2 sin ^{-1}left(x^{2}right)+c ) | 12 |
| 1798 | ( intleft{frac{1}{log x}-frac{1}{(log x)^{2}}right} d x ) | 12 |
| 1799 | If ( c ) is an arbitrary constant then ( int frac{cos (x+a)}{sin (x+b)} d x= ) A ( cdot cos (a-b) ln |sin (x-b)|-x sin (a-b)+c ) B. ( cos (a-b) ln |sin (x+b)|-x sin (a-b)+c ) ( mathbf{c} cdot cos (a+b) ln |sin (x+b)|-x sin (a+b)+c ) D. ( cos (a-b) ln sin |(x+b)|-x sin (a+b)+c ) E ( cdot cos (a-b) ln |sin (x+b)|+x sin (a-b)+c ) | 12 |
| 1800 | ( frac{x}{sqrt{x+4}}, x>0 ) | 12 |
| 1801 | What is ( int_{1}^{e} x ln x d x ) equal to? ( ^{mathrm{A}} cdot frac{e+1}{4} ) B. ( frac{e^{2}+1}{4} ) c. ( frac{e-1}{4} ) D. ( frac{e^{2}-1}{4} ) | 12 |
| 1802 | ( int_{0}^{k} frac{1}{2+8 x^{2}} d x=frac{pi}{16}, ) find the value of ( K ) | 12 |
| 1803 | If ( frac{x}{(x-3)(x-2)}=frac{3}{x-3}+frac{A}{x-2} ) then ( A= ) A . 1 B . 2 ( c cdot-1 ) D. – 2 | 12 |
| 1804 | ( boldsymbol{I}=int_{0}^{1} x^{2} e^{-x} d x ) A ( cdot_{I=1-frac{1}{e}} ) B ( cdot quad I=2-frac{1}{e} ) C ( cdot quad I=2-frac{1}{2 e} ) D ( quad I=2-frac{1}{e^{2}} ) | 12 |
| 1805 | If ( f(y)=e^{y} ) and ( g(y)=y, y>0 ) and ( boldsymbol{F}(boldsymbol{t})=int_{0}^{t} boldsymbol{f}(boldsymbol{t}-boldsymbol{y}) boldsymbol{g}(boldsymbol{y}) d boldsymbol{y} ) then A ( cdot F(t)=e^{t}-(1+t) ) B ( cdot F(t)=t e^{t} ) ( mathbf{c} cdot F(t)=t e^{-t} ) ( mathbf{D} cdot F(t)=1-e^{t}(1+t) ) | 12 |
| 1806 | The value of ( int_{0}^{frac{pi}{2}} log left(frac{4+3 sin x}{4+3 cos x}right) d x ) is ( A cdot 2 ) B. ( frac{3}{4} ) ( c cdot 0 ) D. -2 | 12 |
| 1807 | Integrate ( : int x sin ^{2} x ) | 12 |
| 1808 | Integrate the function ( frac{x cos ^{-1} x}{sqrt{1-x^{2}}} ) | 12 |
| 1809 | Assertion ( int_{0}^{pi / 2} x cot x d x=frac{pi}{2} log 2 ) Reason ( int_{0}^{pi / 2} log sin x d x=-frac{pi}{2} log 2 ) A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion B. Both Assertion & Reason are individually true but Reason is not the , correct (proper) explanation of Assertion c. Assertion is true but Reason is false D. Assertion is false but Reason is true | 12 |
| 1810 | Evaluate ( int sqrt{1+t^{3}} d t ) | 12 |
| 1811 | 19. Jx3 dx is equal to… | 12 |
| 1812 | Solve ( int_{frac{pi}{4}}^{frac{pi}{2}}(2 sin x-cos 5 x) d x ) | 12 |
| 1813 | ( int_{1}^{2} frac{mathbf{d x}}{sqrt{1+mathbf{x}^{2}}}= ) ( ^{mathbf{A}} cdot log _{mathbf{e}}left(frac{2+sqrt{5}}{sqrt{2}+1}right) ) в. ( log _{e}left(frac{sqrt{2}+1}{2+sqrt{5}}right) ) ( ^{mathrm{c}} cdot log _{mathrm{e}}left(frac{2-sqrt{5}}{sqrt{2}-1}right) ) D. | 12 |
| 1814 | Evaluate: ( int frac{x^{3}}{sqrt{1+x^{2}}} d x ) | 12 |
| 1815 | ( n stackrel{L t}{rightarrow} infty ) ( left{frac{sqrt{mathbf{1}}+sqrt{mathbf{2}}+sqrt{mathbf{3}}+ldots+sqrt{boldsymbol{n}}}{boldsymbol{n} sqrt{boldsymbol{n}}}right}= ) ( A cdot O ) B. ( c cdot 2 / 3 ) D. 3/2 | 12 |
| 1816 | Solve ( int frac{1}{sqrt{9-25 x^{2}}} d x ) A. ( frac{1}{5} sin ^{-1}left(frac{5 x}{3}right)+C ) B ( cdot sin ^{-1}left(frac{5 x}{3}right)+C ) c. ( frac{1}{5} sin ^{-1}left(frac{3 x}{5}right)+C ) D. ( sin ^{-1}left(frac{3 x}{5}right)+C ) | 12 |
| 1817 | ( int sqrt{boldsymbol{x}} cdot log boldsymbol{x} boldsymbol{d} boldsymbol{x}= ) A. ( frac{2}{3} x^{3 / 2} cdot log x-frac{4}{9} x^{3 / 2}+c ) B ( cdot frac{2}{3} x^{3 / 2} cdot log x+x^{3 / 2}+c ) c. ( quad x^{3 / 2} cdotleft(log x-frac{2}{3}right)+c ) D ( cdot frac{2}{5} x^{3 / 2}(log x+1)+c ) | 12 |
| 1818 | ( operatorname{Let} boldsymbol{F}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}left(frac{1}{boldsymbol{x}}right) ) where ( f(x)=int_{1}^{x} frac{log t}{1+t} d t ) Then ( F(e) ) is equal to? ( A ) B. 2 ( c cdot 1 / 2 ) ( D ) | 12 |
| 1819 | Assertion Statement-1: ( int frac{x^{2}-1}{left(x^{2}+1right) sqrt{x^{4}+1}} d x= ) ( sec ^{-1}left|frac{x^{2}+1}{x sqrt{2}}right|+C ) Reason staement-2: ( int frac{boldsymbol{d t}}{boldsymbol{t} sqrt{boldsymbol{t}^{2}-boldsymbol{a}}}= ) ( frac{1}{sqrt{a}} sec ^{-1}left|frac{t}{sqrt{a}}right|+C ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1820 | Find ( int frac{sin ^{6} x}{cos ^{8} x} d x ) | 12 |
| 1821 | Solve: ( int frac{sqrt{1+x^{2}}}{x^{4}} d x ) | 12 |
| 1822 | Evaluate ( int_{1}^{2} frac{-1}{x^{2}} d x ) | 12 |
| 1823 | The value of ( int frac{ln nleft(1-left(frac{1}{x}right)right) d x}{x(x-1)} ) is A ( cdot frac{1}{2}left[mleft(1-frac{1}{x}right)^{2}right]+C ) B ( cdot frac{1}{2}left[mleft(1+frac{1}{x}right)right]^{2}+C ) c. ( frac{1}{2} ln (x(x-1))+C ) D ( cdot frac{1}{2}[ln (x(x-1))]^{2}+C ) | 12 |
| 1824 | Evaluate: ( int_{2}^{3} frac{1}{x} d x ) | 12 |
| 1825 | If ( A^{prime} ) s income is ( 30 % ) less than ( B^{prime} s ) then how much per cent is ( B^{prime} ) s income more than ( A^{prime} ) s? A ( 42 frac{6}{7} % ) в. ( 32 frac{1}{10} % ) c. ( 30 % ) D. ( 40 % ) | 12 |
| 1826 | ( int(tan x-cot x)^{2} d x= ) A ( cdot tan x+x+c ) B. ( tan x-x+c ) c. ( tan x-cot x+c ) D. ( tan x-cot x-4 x+c c ) | 12 |
| 1827 | Evaluate ( : int frac{d x}{1+cos a cos x} ) | 12 |
| 1828 | Write an anti derivative for each of the following functions using the method of inspection: i) ( cos 2 x ) ii) ( 3 x^{2}+4 x^{3} ) iii) ( frac{1}{x}, x neq 0 ) | 12 |
| 1829 | Illustration 2.39 Integrate the following w.r.t. x. 1. r 2. x _ 1 3. 2 + 1 / 2 4. _1 2x+3 5. cos (4x +3) 6. cos x | 12 |
| 1830 | Write a value of ( int sqrt{9+x^{2}} d x ) | 12 |
| 1831 | If ( boldsymbol{f}(boldsymbol{x})=int_{0}^{x}(cos (sin t)+cos (cos t) d t ) then ( f(x+pi) ) is? ( mathbf{A} cdot=f(pi)+2 fleft(frac{pi}{2}right) ) B. ( =f(pi)+6 fleft(frac{pi}{6}right) ) ( mathbf{c} cdot=f(pi)+9 fleft(frac{pi}{11}right) ) ( mathbf{D} cdot=f(pi)+10 fleft(frac{pi}{3}right) ) | 12 |
| 1832 | If ( int log (sqrt{1-x}+sqrt{1+x}) d x= ) ( boldsymbol{x} boldsymbol{f}(boldsymbol{x})+boldsymbol{A} boldsymbol{x}+boldsymbol{B} sin ^{-1} boldsymbol{x}+boldsymbol{c}, ) then A ( . f(x)=log (sqrt{1-x}+sqrt{1+x}) ) B. ( _{A}=-frac{1}{3} ) ( c cdot_{B}=frac{2}{3} ) D. ( _{B}=-frac{1}{2} ) | 12 |
| 1833 | Evaluate ( int_{0}^{pi / 4} sin ^{3} 2 t cos 2 t d t ) | 12 |
| 1834 | The value of ( int frac{d x}{xleft(x^{n}+1right)} ) is A ( cdot frac{1}{n} log left(frac{x^{n}}{x^{n}+1}right)+C ) в. ( log left(frac{x^{n}+1}{x^{n}}right)+C ) c. ( frac{1}{n} log left(frac{x^{n}+1}{x^{n}}right)+C ) D. ( log left(frac{x^{n}}{x^{n}+1}right)+C ) | 12 |
| 1835 | ( lim _{n rightarrow infty} frac{left(1^{2}+2^{2}+3^{2}+ldots+n^{2}right)left(1^{3}+2^{3}+right.}{left(1^{6}+2^{6}+3^{6}+ldots+n^{6}right)} ) ( ? ) A ( cdot frac{1}{6} ) B. ( frac{1}{12} ) c. ( frac{7}{12} ) D. ( frac{1}{7} ) | 12 |
| 1836 | Evaluate ( int_{0}^{2 pi} frac{x sin ^{2 n} x}{sin ^{2 n} x+cos ^{2 n} x} d x, ) for ( boldsymbol{n}>mathbf{0} ) ( A ) B. ( 2 pi ) ( mathbf{c} cdot pi^{2} ) D. ( frac{1}{2} pi ) | 12 |
| 1837 | ( int_{0}^{pi} frac{x}{a^{2} cos ^{2} x+b^{2} sin ^{2} x} d x ) | 12 |
| 1838 | Evaluate: ( int_{-a}^{a} frac{sqrt{a-x}}{sqrt{a+x}} d x ) | 12 |
| 1839 | The value of ( intleft(x e^{ell n sin x}-cos xright) d x ) is equal to: ( mathbf{A} cdot x cos x+C ) B. ( sin x-x cos +C ) c. ( -e^{e n x} cos x+C ) ( mathbf{D} cdot sin x+x cos x+C ) | 12 |
| 1840 | Find the values of ( c ) that satisfy the Rolle’s theorem for integrals on [-2,1] ( boldsymbol{f}(boldsymbol{t})=boldsymbol{2} boldsymbol{t}-boldsymbol{t}^{3}-boldsymbol{t}^{2} ) This question has multiple correct options ( ^{mathrm{A}} cdot_{c}=frac{1+sqrt{7}}{-3} ) B. ( c=frac{1-sqrt{7}}{-3} ) ( c_{c}=frac{-1+sqrt{7}}{-3} ) D. ( c=frac{-1-sqrt{7}}{-3} ) | 12 |
| 1841 | The antiderivative of ( frac{x+left(cos ^{-1} 3 xright)^{2}}{sqrt{1-9 x^{2}}} ) is A ( cdot C-frac{1}{9}left[sqrt{1-9 x^{2}}+left(cos ^{-1} 3 xright)^{3}right] ) B. ( C+frac{1}{9}left[sqrt{1-9 x^{2}}+left(cos ^{-1} 3 xright)^{2}right. ) c. ( c-frac{1}{3}left[left(1-9 x^{2}right)^{3 / 2}+left(cos ^{-1} 3 xright)^{3}right. ) D. none of these | 12 |
| 1842 | Evaluate: ( int frac{1}{a^{x} b^{x}} d x ) | 12 |
| 1843 | Evaluate ( int_{0}^{1}left(2 x^{2}+x+1right) d x ) | 12 |
| 1844 | Evaluate the integral ( int_{0}^{infty} e^{-2 x} cdot sin 5 x d x ) A ( cdot frac{-2}{29} ) в. ( frac{2}{29} ) c. ( frac{5}{29} ) D. ( frac{7}{25} ) | 12 |
| 1845 | ( frac{boldsymbol{x}^{2}+mathbf{5} boldsymbol{x}+mathbf{7}}{(boldsymbol{x}-mathbf{3})^{3}}=frac{boldsymbol{A}}{boldsymbol{x}-mathbf{3}}+frac{boldsymbol{B}}{(boldsymbol{x}-mathbf{3})^{2}}+ ) ( frac{C}{(x-3)^{3}} Rightarrow A= ) ( A cdot 2 ) B . – – ( c ) ( D ) | 12 |
| 1846 | Assertion If ( a>0 ) and ( b^{2}-4 a c0, b^{2}-4 a c<0, text { then } a x^{2}+b x+right. ) ( c ) can be written as sum of two squares. A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1847 | Evaluate ( boldsymbol{I}=int_{boldsymbol{pi} / mathbf{6}}^{boldsymbol{pi} / mathbf{3}} sin boldsymbol{x} boldsymbol{d} boldsymbol{x} ) A ( cdot frac{1-sqrt{3}}{2} ) в. ( frac{sqrt{3}+1}{2} ) c. ( frac{sqrt{3}-1}{2 sqrt{3}} ) D. None of these | 12 |
| 1848 | Find the integral of ( intleft(2 x^{2}-3 sin x+right. ) ( mathbf{5} sqrt{boldsymbol{x}}) boldsymbol{d} boldsymbol{x} ) | 12 |
| 1849 | Evaluate: ( int_{2}^{1}|boldsymbol{x}-mathbf{3}| boldsymbol{d} boldsymbol{x} ) | 12 |
| 1850 | A disc, sliding on an inclined plane, is found to have its position (measured from the top of the plane) at any instant given by ( boldsymbol{x}=mathbf{3} boldsymbol{t}^{2}+mathbf{1} ) where ( boldsymbol{x} ) is in meter and ( t ) in second. Its average velocity in the time interval between 2 s to 2 is ( mathbf{A} cdot 10.2 mathrm{ms}^{-1} ) B . ( 15.5 mathrm{ms}^{-1} ) ( mathbf{c} cdot 12.3 mathrm{ms}^{-1} ) D. ( 9.7 m s^{-1} ) | 12 |
| 1851 | ( int_{0}^{pi} x f(sin x) d x ) equals ( ^{mathbf{A}} cdot_{2 pi} int_{0}^{frac{pi}{2}} f(sin x) d x ) в. ( pi int_{0}^{pi} f(sin x) d x ) c. ( quad pi int_{0}^{frac{pi}{2}} f(sin x) d x ) D. None of these | 12 |
| 1852 | Evaluate ( int frac{1}{sqrt{3} sin x+cos x} d x ) | 12 |
| 1853 | Evaluate the following definite integral: ( int_{0}^{4} 4 x-x^{2} d x ) | 12 |
| 1854 | 2. Let a, b, c be non-zero real numbers such that J (1+cos® x)(ax² + bx +c) dx = ſ(1+cos® x)(ar? + bx + c) dx. Then the quadratic equation ax2 +bx+c = 0 has (1981 – 2 Marks) (a) no root in (0,2) (b) at least one root in (0,2) (c) a double root in (0,2) (d) two imaginary roots | 12 |
| 1855 | If ( frac{mathbf{3} boldsymbol{x}+boldsymbol{a}}{boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+mathbf{2}}=frac{boldsymbol{A}}{boldsymbol{x}-mathbf{2}}-frac{mathbf{1 0}}{boldsymbol{x}-mathbf{1}}, ) then ( boldsymbol{a}=ldots ) and ( boldsymbol{A}=ldots ) A ( cdot a=7, A=13 ) 3 В. ( a=11, A=13 ) c. ( a=13, A=7 ) | 12 |
| 1856 | Evaluate the integral ( int_{0}^{1}left(1-x^{2}right) d x ) | 12 |
| 1857 | Solve ( : int sec ^{-1} sqrt{x} d x ) | 12 |
| 1858 | Find the values of ( c ) that satisfy the Rolle’s theorem for integrals on [-1,3] ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}-boldsymbol{8} ) ( mathbf{A} cdot c=3 ) в. ( c=1 ) ( mathbf{c} cdot c=0 ) ( mathbf{D} cdot c=2 ) | 12 |
| 1859 | Find the area bounded by the ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) and the ordinates ( x=0 ) and ( x=a e, ) where ( b^{2}=a^{2}left(1-e^{2}right) ) and ( e<1 ) | 12 |
| 1860 | 13. Evaluate sin 1 2x+2 2x+2 dx 4x² + 8x+13) | 12 |
| 1861 | Evaluate the integral ( int_{3}^{5}(2-x) d x ) | 12 |
| 1862 | Find ( intleft(x^{2}+1right) d x ) | 12 |
| 1863 | Evaluate the integral ( int_{0}^{frac{pi}{2}} frac{sin ^{frac{3}{2} x} d x}{sin ^{frac{3}{2}} x+cos ^{frac{3}{2}} x} ) A ( cdot frac{pi}{2} ) в. ( frac{pi}{4} ) c. ( pi ) D. | 12 |
| 1864 | Evaluate ( int frac{boldsymbol{d x}}{sqrt{(boldsymbol{x}-boldsymbol{a})(boldsymbol{b}-boldsymbol{x})}} ) A ( cdot I=2 sin ^{-1} sqrt{frac{x-a}{(b-a)}}+C ) в. ( I=2 cos ^{-1} sqrt{frac{x-a}{(b-a)}}+C ) ( ^{mathrm{c}} cdot_{I=sin ^{-1}} sqrt{frac{x-a}{(b-a)}}+C ) D. ( _{I}=2 sin ^{-1} sqrt{frac{x-b}{(a-b)}}+C ) | 12 |
| 1865 | ( intleft(3 x^{2}+2 xright) d x ) | 12 |
| 1866 | Solve ( int frac{1}{x^{5}}left(1+x^{4}right) d x ) | 12 |
| 1867 | Solve: ( int e^{x}left(tan ^{-1} x+frac{1}{1+x^{2}}right) d x ) A ( cdot e^{x} tan ^{-1} x+c ) B. ( frac{e^{x}}{1+x^{2}}+c ) ( mathbf{c} cdot e^{x} tan x+c ) D. None of these | 12 |
| 1868 | ( int_{0}^{2 pi} frac{x sin ^{2 n} x}{sin ^{2 n} x+cos ^{2 n} x} d x ) ( mathbf{A} cdot pi^{2} ) B . ( 2 pi^{2} ) ( mathbf{c} cdot 4 pi^{2} ) D. ( 8 pi^{2} ) | 12 |
| 1869 | If ( int frac{2 sin x+3 cos x}{3 sin x+4 cos x} d x=A log ) ( |3 sin x+4 cos x|+B x+c, ) then ( A= ) ( ldots ldots ldots, B=ldots ldots ldots . . ) A. ( -frac{1}{25}, frac{18}{25} ) 8. ( -frac{1}{5},-frac{1}{5} ) c. ( frac{1}{25}, frac{18}{25} ) D. ( frac{1}{25}, frac{3}{25} ) | 12 |
| 1870 | The average ordinate of ( y=sin x ) over the interval ( [mathbf{0}, boldsymbol{pi}] ) is – A. ( 1 / pi ) B. ( 2 / pi ) c. ( 4 / pi^{2} ) D. ( 2 / pi^{2} ) | 12 |
| 1871 | ( int frac{sin (2 x)}{1+cos ^{2} x} d x ) is equal to A ( cdot-frac{1}{2} log left(1+cos ^{2} xright)+c ) B . ( 2 log left(1+cos ^{2} xright)+c ) c. ( frac{1}{2} log (1+cos 2 x)+c ) D. ( c-log left(1+cos ^{2} xright) ) | 12 |
| 1872 | ( int x cos ^{2} 2 x d x ) | 12 |
| 1873 | If ( int frac{x cos ^{-1} x}{sqrt{1-x^{2}}} d x= ) ( -kleft[sqrt{1-x^{2}} cos ^{-1} x+xright]+C . ) what will be the value of ( k ? ) | 12 |
| 1874 | Evaluate ( int frac{cos ^{2} x}{sin ^{3} xleft(sin ^{5} x+cos ^{5} xright)^{frac{3}{5}}} d x ) | 12 |
| 1875 | ntegrate the function ( frac{x+2}{sqrt{x^{2}+2 x+3}} ) | 12 |
| 1876 | ( int frac{x^{2}-1}{x^{3} sqrt{2 x^{4}-2 x^{2}+1}} d x ) is equal to A ( cdot frac{sqrt{2 mathrm{x}^{4}-2 mathrm{x}^{2}+1}}{mathrm{x}^{2}}+mathrm{c} ) в. ( frac{sqrt{2 mathrm{x}^{4}-2 mathrm{x}^{2}+1}}{mathrm{x}^{3}}+mathrm{c} ) c. ( frac{sqrt{2 x^{4}-2 x^{2}+1}}{x}+c ) D. ( frac{sqrt{2 mathrm{x}^{4}-2 mathrm{x}^{2}+1}}{2 mathrm{x}^{2}}+mathrm{c} ) | 12 |
| 1877 | Write a value of ( int e^{x}left(frac{1}{x}-frac{1}{x^{2}}right) d x ) | 12 |
| 1878 | Evaluate: ( int_{0}^{frac{pi}{2}} frac{sin x-cos x}{1+sin x cos x} d x ) | 12 |
| 1879 | Solve: ( int frac{log x}{x^{2}} d x ) | 12 |
| 1880 | Assertion ( int_{0}^{pi / 4} frac{cos x+sin x}{cos ^{2} x+sin ^{4} x} d x=frac{pi}{4}+ ) ( frac{1}{2 sqrt{3}} log (2+sqrt{3})=I ) Reason ( boldsymbol{I}=int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{1-boldsymbol{x}^{2}+boldsymbol{x}^{4}} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 12 |
| 1881 | The value of ( lim _{n rightarrow infty} Sigma_{i=1}^{n-1} sqrt{4+frac{5 i}{n}} ) is equal to? A ( cdot 15 / 38 ) B. ( 38 / 15 ) c. ( 21 / 15 ) D . ( 22 / 15 ) | 12 |
| 1882 | Evaluate ( int_{a}^{b} x sin x d x ) | 12 |
| 1883 | ( int_{0}^{frac{2}{3}} frac{d x}{4+9 x^{2}}= ) | 12 |
| 1884 | If ( frac{mathbf{3} boldsymbol{x}^{2}+mathbf{1 0 x}+mathbf{1 3}}{(boldsymbol{x}-mathbf{1})^{4}}=frac{boldsymbol{A}}{(boldsymbol{x}-mathbf{1})^{2}}+ ) ( frac{B}{(x-1)^{3}}+frac{C}{(x-1)^{4}} ) then descending order of ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) A. ( A, B, C ) в. ( C, B, A ) c. ( A, C, B ) D. ( C, A, B ) | 12 |
| 1885 | ( int cos x cdot cos 2 x cdot cos 3 x d x ) | 12 |
| 1886 | ( int_{0}^{a} frac{x-a}{x+a} d x= ) ( mathbf{A} cdot a+2 a log 2 ) B. ( a-2 a log 2 ) c. ( 2 a log -a ) D. ( 2 a log 2 ) | 12 |
| 1887 | ( int frac{boldsymbol{x}^{2}}{boldsymbol{x}^{6}+boldsymbol{2} boldsymbol{x}^{3}-boldsymbol{3}} boldsymbol{d} boldsymbol{x}= ) A. ( frac{1}{12} log left|frac{x^{3}-1}{x^{3}+1}right|+c ) в. ( frac{1}{12} log left|frac{x^{3}-1}{x^{3}+3}right|+c ) c. ( frac{1}{12} log left|frac{x^{3}+3}{x^{3}-1}right|+c ) D. ( frac{1}{12} log left|frac{x^{3}-3}{x^{3}+1}right|+c ) | 12 |
| 1888 | If ( int frac{2 d x}{[(x-5)+(x-7)] sqrt{(x-5)(x-7)}} ) ( boldsymbol{f}[boldsymbol{g}(boldsymbol{x})]+boldsymbol{c}, ) then A ( cdot f(x)=sin ^{-1} x, g(x)=sqrt{(x-5)(x-7)} ) B . ( f(x)=sin ^{-1} x, g(x)=(x-5)(x-7) ) C ( cdot f(x)=tan ^{-1} x, g(x)=sqrt{(x-5)(x-7)} ) D. ( f(x)=tan ^{-1} x, g(x)=(x-5)(x-7) ) | 12 |
| 1889 | ( int sqrt{e^{x}+1} d x ) | 12 |
| 1890 | Number of real solution of the given equation for ( x, int x^{2} e^{x} d x=0 ) | 12 |
| 1891 | Evaluate the integral ( int_{1}^{4}left(x^{2}-xright) d x ) | 12 |
| 1892 | ( mathbf{f} boldsymbol{y}=int frac{boldsymbol{d} boldsymbol{x}}{left(mathbf{1}+boldsymbol{x}^{2}right)^{3 / 2}} ) and ( boldsymbol{y}=mathbf{0} ) when ( boldsymbol{x}=mathbf{0}, ) then value of ( mathbf{y} ) when ( boldsymbol{x}=mathbf{1}, ) is | 12 |
| 1893 | Evaluate ( int frac{boldsymbol{x}+mathbf{9}}{(boldsymbol{x}+mathbf{1 0})^{2}} boldsymbol{e}^{x} boldsymbol{d} boldsymbol{x}= ) A ( cdot quad e^{x} frac{1}{x+9}+c ) B ( cdot e^{x} frac{1}{x+10}+c ) c. ( quad e^{x} frac{1}{(x+9)^{2}}+c ) D. ( e^{x}+c frac{1}{(x+10)^{2}} ) | 12 |
| 1894 | ( int frac{1+2 x^{2}}{x^{2}left(1+x^{2}right)} d x= ) A. ( quad tan ^{-1} x+frac{1}{x}+c ) B. ( tan ^{-1} x-frac{1}{x}+c ) c. ( frac{tan ^{-1} x}{x}+c ) D. ( frac{tan ^{-1} x}{x^{2}}+c ) | 12 |
| 1895 | If ( f(x)=int_{0}^{x} t sin t d t, ) then ( f^{prime}(x) ) is A ( cdot cos x+x sin x ) B. ( x sin x ) c. ( x cos x ) ( mathbf{D} cdot sin x+x cos x ) | 12 |
| 1896 | ( lim _{n rightarrow infty} nleft[frac{1}{(n+1)(n+2)}+frac{1}{(n+2)(n+}right. ) is equal to ( ^{A} cdot log left(frac{3}{2}right) ) в. ( log left(frac{5}{2}right) ) c. ( log left(frac{1}{2}right) ) D. ( log left(frac{7}{4}right) ) | 12 |
| 1897 | ( int_{0}^{2 pi} sin ^{4} x d x ) is equal to This question has multiple correct options ( mathbf{A} cdot 2 int_{0}^{pi} sin ^{4} x d c ) B. ( 8 int_{0}^{frac{pi}{4}} sin ^{4} x d x ) C ( cdot 4 int_{0}^{frac{pi}{2}} cos ^{4} x d x ) D. ( 3 int_{0}^{frac{2 pi}{3}} sin ^{4} x d x ) | 12 |
| 1898 | What is ( int(x cos x+sin x) d x ) equal to? Where ( c ) is an arbitrary constant ( mathbf{A} cdot x sin x+c ) B. ( x cos x+c ) ( c cdot-x sin x+c ) D. ( -x cos x+c ) | 12 |
| 1899 | Find the partial fraction ( frac{2 x+1}{(3 x+2)left(4 x^{2}+5 x+6right)} ) | 12 |
| 1900 | Evaluate: ( int(boldsymbol{P}+boldsymbol{Q}) boldsymbol{d} boldsymbol{x} ) ( mathbf{A} cdot int P d x+int Q d x ) в. ( int P d x+Q+C ) c. ( int P d x-int Q d x ) D. None of the above | 12 |
| 1901 | ( int frac{1}{sqrt{1-4 x^{2}}} d x= ) ( A cdot sin ^{-1} 2 x+c ) B. ( frac{1}{2} sin ^{-1} 2 x+c ) c. ( left(sin ^{-1} 2 xright)^{2}+c ) D. ( frac{1}{2}left(sin ^{-1} 2 xright)^{2}+c ) | 12 |
| 1902 | show jus cinse) dr = Frosine) ds. (1982 – 2 Mai | 12 |
| 1903 | The value of ( int_{0}^{frac{pi}{4}}(sqrt{tan x}+sqrt{cot x}) d x ) is equal to A ( cdot frac{pi}{2} ) B. ( -frac{pi}{2} ) c. ( frac{pi}{sqrt{2}} ) D. ( -frac{pi}{sqrt{2}} ) | 12 |
| 1904 | ( int frac{2 x+5}{x^{2}+5 x-3} d x ) | 12 |
| 1905 | If ( int frac{d x}{(x+2)left(x^{2}+1right)}=a ln left(1+x^{2}right)+ ) ( b tan ^{-1} x+frac{1}{5} ln |x+2|+C ) then A. ( a=-frac{1}{10}, b=-frac{2}{5} ) B. ( a=frac{1}{10}, b=frac{2}{5} ) c. ( a=-frac{1}{10}, b=frac{2}{5} ) D. ( a=frac{1}{10}, b=-frac{2}{5} ) | 12 |
| 1906 | Evaluate the given integral. ( int frac{2}{1-cos 2 x} d x ) | 12 |
| 1907 | 30. Evaluate the definite integral : – 1/3 (1-x² | 12 |
| 1908 | Integrate the rational function ( frac{x^{3}+x+1}{x^{2}-1} ) | 12 |
| 1909 | The number of partial fraction of ( frac{3 x^{2}+70 x+93}{(x-1)^{4}} ) is A . 3 B. 4 ( c .5 ) D. 2 | 12 |
| 1910 | If ( I_{n}=int_{0}^{pi / 4} tan ^{n} x times sec ^{2} x d x, ) then ( boldsymbol{I}_{1}, boldsymbol{I}_{2}, boldsymbol{I}_{3}, ldots . . . ) are in A. A.P в. G. c. н.P D. none | 12 |
| 1911 | Find ( : int log x cdot d x ) | 12 |
| 1912 | Evaluate the given integral. ( int x sin 2 x d x ) | 12 |
| 1913 | Evaluate the definite integral ( int_{0}^{frac{pi}{2}} cos 2 x d x ) | 12 |
| 1914 | ( boldsymbol{I}=int_{0}^{1} frac{(1-boldsymbol{x}) boldsymbol{d} boldsymbol{x}}{(mathbf{1}+boldsymbol{x})} ) | 12 |
| 1915 | For what ( a<0 ) does the inequality ( int_{a}^{0}left(3^{-2 x} 2.3^{-x}right) d x geqslant 0 ) hold true? | 12 |
| 1916 | ( int_{0}^{1} frac{x e^{x}}{(x+1)^{2}} d x= ) A. ( frac{e}{2} ) B. ( frac{e-1}{2} ) c. ( frac{e}{2}-1 ) D. ( frac{e-3}{2} ) | 12 |
| 1917 | Let ( f ) be a polynomial function such that ( boldsymbol{f}(mathbf{3} boldsymbol{x})=boldsymbol{f}^{prime}(boldsymbol{x}) cdot boldsymbol{f}^{prime prime}(boldsymbol{x}), ) for all ( boldsymbol{x} in ) ( boldsymbol{R} ). Then: A ( cdot f(2)+f^{prime}(2)=28 ) B . ( f^{prime prime}(2)-f^{prime}(2)=0 ) ( mathbf{c} cdot f^{prime prime}(2)-f(2)=4 ) D ( cdot f(2)-f^{prime}(2)+f^{prime prime}(2)=10 ) | 12 |
| 1918 | If ( frac{x}{(x-3)(x-2)}=frac{3}{x-3}+frac{A}{x-2} ) then ( A= ) A . 1 B . 2 ( c cdot-1 ) D. – 2 | 12 |
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