Integrals Questions

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
A. An indefinite integral of a function fis a differentiable function F whose derivative is equal to the original function f.
B. This can be stated symbolically as ( F=f )
c. without upper and lower limits, also called an antiderivative
D. None of the above

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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