# 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 NoQuestionsClass
1Evaluate: ( 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
3Illustration 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
5Write a value of
( int frac{sin x-cos x}{sqrt{1+sin 2 x}} d x )
12
6Solve: ( int frac{sin x-cos x}{sqrt{sin 2 x}} d x )12
7Evaluate ( 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
8By 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
9Find: ( 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
127. Evaluate: scos20″ dat
Evaluate :
sin x
12
138. 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
14Resolve ( 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
15sin 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
18Prove ( 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
19Resolve 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
20The 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}}} ) is12
21Solve:- ( sin ^{-1}(cos x) )12
22Let ( 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
23Integrate ( 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
2510. 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
29Evaluate ( 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
33If ( 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
36Solve :
( boldsymbol{I}=int sin ^{6} boldsymbol{x} boldsymbol{d} boldsymbol{x} )
12
37If ( 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
39Evaluate ( 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
41Evaluate the following integrals:
( int frac{1}{sqrt{7-3 x-2 x^{2}}} d x )
12
42Evaluate 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
46Evaluate: ( int frac{(x(pi+49))^{15 / 7}}{pi^{2}left(x^{pi}+7right)} d x )12
47Integrate 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
49For 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
51Evaluate: ( 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
52Evaluate ( int frac{1}{a^{x} b^{x}} d x )12
53The 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
54Find the integral of the function ( frac{cos x-sin x}{1+sin 2 x} )12
55The 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
56Evaluate ( 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
57Find 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
58The 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
59Evaluate: ( 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
61The 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
62Let ( 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
64The 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
65Evaluate ( int e^{x} sin e^{x} d x ) on ( R )12
66Resolve ( 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
67The 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
68Evaluate the definite integral:
( int_{0}^{pi / 2} sin x d x )
12
69Integrate :
( intleft(x^{4}-x^{2}+1-frac{2}{1+x^{2}}right) d x )
12
70Find the integral of ( intleft(2 x^{2}-3 sin x+right. ) ( mathbf{5} sqrt{boldsymbol{x}}) boldsymbol{d} boldsymbol{x} )12
71If ( 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
72Resolve 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
73Evaluate :
( int_{0}^{pi} frac{x tan x}{sec x+tan x} )
12
74Evaluate the integral ( int_{0}^{1 / 2} frac{x sin ^{-1} x}{sqrt{1-x^{2}}} d x )12
75Evaluate: ( intleft(sqrt{boldsymbol{x}}+frac{mathbf{1}}{sqrt{boldsymbol{x}}}right) boldsymbol{d} boldsymbol{x} )12
76Solve:
( 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
77Evaluate:
( 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
79Evaluate:
( int frac{sin x-cos x}{sqrt{sin 2 x}} d x )
12
80Show 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
81Evaluate the given integral. ( int e^{x}left(frac{1+x}{(2+x)^{2}}right) d x )12
82The 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
84If ( 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
85Evaluate 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
87Solve ( : 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
89Find ( 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
9126. 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
92Evaluate the given integral: ( int_{0}^{1}(1+ )
( x)^{5} d x )
12
93Evaluate :
( 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
95If ( 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
96Solve ( : int_{0}^{1} e^{e^{x}}left(1+x cdot e^{x}right) d x )12
97Resolve 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
99If 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
100Evaluate :
( 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
103Solve:
( int_{frac{pi}{6}}^{frac{pi}{3}} frac{sin x+cos x}{sqrt{sin 2 x}} d x )
12
104Solve : ( int e^{x} cdot sin 3 x d x )12
105If ( 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
106Evaluate the following integral:
( int_{0}^{pi} x d x )
12
107If ( 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
108Integrate the rational function
( frac{3 x+5}{x^{3}-x^{2}-x+1} )
12
109Evaluate: ( int sqrt{tan x} d x,left(0<x<frac{pi}{2}right) )12
110Evaluate : ( 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
111The 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
113Integrate 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
115The 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
116If ( 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
117Evaluate ( 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
122Integrate 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
12412. Integrateſ xº+3x+2_dx.
+
2
12
125If 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
127Suppose ( 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
128Evaluate the following:
( int(3 x+1) sqrt{2 x-1} d x )
12
129Assertion
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
130Integrate: ( 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
132if ( 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
133Number 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
134Solve: ( int frac{boldsymbol{d x}}{boldsymbol{x}left(boldsymbol{a}+boldsymbol{b} boldsymbol{x}^{n}right)^{2}} )12
135Evaluate 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
136Derive 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
138If ( 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
143If ( 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
144Evaluate: ( int frac{cos 2 x}{sin x} d x )12
145dx
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
147Obtain ( 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
149Evaluate ( 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
152Evaluate ( 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
153Let ( 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
1543.
| (x ² ax is
(a) 2-2
(c) V2 – 1
(6) 2+ √2
(d) -√2-√3+5
12
155Find ( 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
157Evaluate :
( boldsymbol{I}=int frac{boldsymbol{2} boldsymbol{x}}{boldsymbol{x}^{2}-boldsymbol{6} boldsymbol{x}+boldsymbol{6}} boldsymbol{d} boldsymbol{x} )
12
158If ( 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
159Evaluate ( 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
160If ( 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
161Evalaute 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
162The 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
163Integrate:
( int frac{boldsymbol{v}}{1-boldsymbol{v}}= )
12
164f(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
165Evaluate: ( 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
166Assertion
( 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
168Evaluate ( int_{0}^{2} frac{6 x+3}{x^{2}+4} d x )12
169If ( 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
170Evaluate: ( int_{0}^{pi / 8} cos ^{3} 4 x d x )
A . ( 1 / 6 )
в. ( 1 / 5 )
c. ( -1 / 3 )
D. 1/
12
171If ( 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
173Evaluate
( 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
175Solve
( int frac{v}{1-v} d v )
12
176Evaluate:
( 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
180If ( 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
181Integrate the following function.
( sin x sin (cos x) )
12
182Evaluate: ( int_{0}^{1} frac{sqrt{tan ^{-1} x}}{1+x^{2}} d x )12
183The 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
185Evaluate:
( 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
187The 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
188Assertion
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
189ntegrate 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
191Evaluate 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
193Solve:
( 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
195Evaluate the definite integral:
( int_{0}^{pi / 2} cos x d x )
12
196Evaluate ( int_{0}^{1} e^{2-3 x} d x ) as a limit of a sum.12
197Evaluate the following integral:
( int frac{e^{x}}{sqrt{16-e^{2 x}}} d x )
12
198Evaluate
( int frac{(log x)^{2}}{x} d x )
12
199Find ( 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
201If ( 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
203Evaluate the following integral:
( int frac{sin 2 x}{(a+b cos 2 x)^{2}} d x )
12
204The 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
208The 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
2097.
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
210Evaluate ( 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
211Evaluate ( int frac{1}{left(e^{x}-1right)} d x )12
212Evaluate 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
214Find 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
216Evaluate the following integral:
( int_{0}^{pi / 2} cos x d x )
12
217Solve :
( int_{1}^{2} frac{x}{(x+1)(x+2)} d x )
12
21835. 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
219What 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
220Evaluate the following definite integral:
( int_{0}^{1}left(3 x^{2}+2 xright) d x )
12
2219.
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
223Evaluate ( int_{1}^{2} frac{-1}{x^{2}} d x )12
224Evaluate ( 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
226c
2.
Evaluate / xdx
(a+bx) ²
12
227Integrate:
( int e^{x} sin x cdot d x )
12
228Solve:
( 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
230Verify mean value theorem for the function ( f(x)=x^{2} ) in the interval [2,4]12
231Evaluate: ( 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
234Solve:
( int_{0}^{2} x sqrt{x+2} d x )
12
235Evalute ( int frac{cot x}{sqrt{sin x}} d x )12
236Evaluate the following integral:
( int_{0}^{5} x^{2} d x )
12
237Evaluate 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
241Evaluate ( 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
242Integrate:
( intleft(left(frac{x+1}{x-1}right)^{2}+left(frac{x-1}{x+1}right)^{2}-2right)^{frac{1}{2}} d x )
12
243Evaluate:
( 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
246Integrate the rational function ( frac{2 x}{left(x^{2}+1right)left(x^{2}+3right)} )12
247What 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
250If ( 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
253Solve ( 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
25598
pk+1
17.
1
k +1
-dxthen
x(x+1)
krl Jk
(JEE Adv. 2017)
(b) I log, 99
(c) 1 50
50
12
256Integrate 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
2587.
* 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
260Evaluate ( 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
261Evaluate: ( 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
263State 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
264Integrate the function ( x sin x )12
26510. 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
267If ( 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
269Solve:
( 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
271Evaluate the following integral:
( int_{0}^{3}left(2 x^{2}+3 x+5right) d x )
12
272Solve ( int frac{x^{2}}{x+1} d x )12
273Solve ( : 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
27622.
The value of integra
3 V9-x+ Fax is
(C)
2
(d)
1
12
2771923
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
279The 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
280Express ( int_{0}^{4} x^{3} d x ) as limit of sum and thus evaluate it.12
281Number of Partial Fractions of ( frac{3 x^{2}+1}{left(x^{2}+1right)^{4}} )
A .4
B. 3
( c cdot 2 )
D.
12
282Evaluate: ( 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
283Evaluate ( : int frac{1}{sqrt{x^{3}}} d x )12
284Evaluate: ( int frac{cos x+sin x}{sqrt{sin 2 x}} d x )12
285Integrate:
( int frac{x^{2}+1}{(x+1)^{3}(x-2)} d x )
12
286Evaluate the following integral:
( int_{0}^{2}left(x^{2}+xright) d x )
12
287Evaluate ( : int_{-pi}^{pi} frac{sin ^{2} x}{1+e^{x}} d x )12
288Solve ( int_{boldsymbol{pi} / mathbf{6}}^{boldsymbol{pi} / mathbf{3}} frac{mathbf{1}}{sin mathbf{2} boldsymbol{x}} boldsymbol{d} boldsymbol{x} )12
2892.
2 sin x -sin 2x
f(x) is the integral of –
x #0, find lim
f'(x)
→0
(1979)
12
290Evaluate ( 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
291Evaluate the definite integral:
( int_{1}^{2} frac{1}{x} d x )
12
292Evaluate the given integral.
( int frac{1}{cos x-sin x} d x )
12
293Assertion
( 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
295If ( 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
296Evaluate ( 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
298Evaluate the following integral:
( int frac{1}{sqrt{1+4 x^{2}}} d x )
12
299Evaluate ( int_{0}^{2} 2 x d x )12
300The 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
301Evaluate: ( int_{-2}^{3} frac{1}{x+5} d x )12
302Solve ( 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
304Find ( int frac{2 x}{left(x^{2}+1right)left(x^{4}+4right)} d x )12
305Evaluate ( int frac{2 x}{1+x^{2}} d x )12
306By 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
309The 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
310Evaluate :
( int frac{(1+log x)^{2}}{x} d x )
12
311Evaluate :
( 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
315The function ( f(x)=x^{3}-7 x^{2}+25 x+ )
8 has exactly roots.
( A cdot 2 )
B.
( c .3 )
D.
12
316The 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
317Integrate: ( 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
318Solve:-
( int t sqrt{frac{t^{2}+1}{t^{2}-1}} d t )
12
319The 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
320Evaluate the given integral.
( int frac{1}{x^{4}-1} d x )
12
321Integrate:
( 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
325Evaluate: ( 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
326Find 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
32711-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
328Solve ( int(3 x-2) sqrt{2 x^{2}-x+1} d x )12
329Solve: ( int_{0}^{pi / 4} frac{sin x cos x}{cos ^{4} x+sin ^{4} x} d x )12
330Evaluate the integral ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x )12
331The 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
332Find 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
334Solve : ( int frac{x d x}{left(x^{2}+a^{2}right)left(x^{2}+b^{2}right)} )12
335Evaluate 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
337Evaluate 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
338Evaluate ( 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
340Integrals 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
342Solve: ( int frac{1}{sqrt{1-e^{2 x}}} d x )12
343Evaluate : ( int frac{x^{2}-1}{(x-1)^{2}(x+3)} d x )12
344Evaluate ( int_{0}^{1}left(x+x^{2}right) d x )12
345JL
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
349Show 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
351Evaluate the definite integral ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{1-boldsymbol{x}^{2}}} )12
352Prove 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
354X
+
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
355Evaluate:
( int sin ^{4} x cos ^{4} x d x )
12
356Evaluate ( int_{0}^{2} frac{x}{3} d x )12
357Solve : ( int x^{2}left(1-frac{1}{x^{2}}right) d x )12
358Evaluate the following integral:
( int_{0}^{pi} 1+sin x d x )
12
359ble 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
360Evaluate ( int frac{e^{x-1}+x^{e-1}}{e^{x}+x^{e}} d x )12
361Solve: ( 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
364Find ( int frac{sin x}{sin 4 x} d x )12
36547. The integral
[JEE
– is equal to:
1+ cos x
(b) -2
(d) 4
(a)
(c)
-1
2
12
366Evaluate: ( 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
370Integrate:
( 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
37421. 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
375If ( 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
376The 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
377Evaluate: ( int frac{cos 2 x-cos 2 alpha}{cos x-cos alpha} d x )12
378Show that ( int a^{x} e^{x} d x=frac{a^{x} e^{x}}{log a+1} )12
379let ( 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
380Solve
( int frac{cos (x+a)}{cos (x-a)} d x )
12
381Find 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
383Evaluate ( int frac{boldsymbol{d x}}{sqrt{mathbf{2 a x}-boldsymbol{x}^{2}}} )12
384Solve ( intleft(4 e^{3 x}+1right) d x )12
385The value of integral ( int_{pi / 4}^{pi / 2} cos x d x ) is?12
38616. 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
389Assertion
( 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
391The 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
392Solve :
( 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
394Solve:
( 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
395Evaluate the definite integral:
( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x )
12
396If ( 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
397Integrate 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
399Let ( 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
401Integrate ( int e^{x}left(frac{x^{2}+3 x+3}{(x+2)^{2}}right) d x )12
402Evaluate the following integral ( int frac{x sin ^{-1} x^{2}}{sqrt{1-x^{4}}} d x )12
403By 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
404If ( 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
405Solve: ( int cos ^{2} x sin ^{2} x d x )12
406( boldsymbol{x}^{5} sqrt{boldsymbol{a}^{3}+boldsymbol{x}^{3}} )12
407Evaluate 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
411Evaluate ( 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
414Evaluate the given integral.
( int e^{x}(sec x(1+tan x)) d x )
12
415The 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
418Evaluate the following integral as limit of sum:
( int_{0}^{2} e^{x} d x )
12
419If ( 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
420Let ( 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
423Find: ( 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
425Evaluate 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
427If ( 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
428Evaluate the following integrals:
( int frac{1}{x^{2 / 3} sqrt{x^{2 / 3}-4}} d x )
12
429Evaluate 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
431Prove that:
( int tan ^{3} 2 x sec 2 x d x )
12
432Evaluate ( 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
434Evaluate 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
436Find ( int_{0}^{5}(x+1) d x ) as limit of a sum12
437Let ( 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
439et 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
441Solve :
( int frac{e^{x}}{-2left(1+e^{-x}right)^{2}} cdot d x )
12
442Solve 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
444Evaluate: ( int frac{1}{sin x+sec x} d x )12
445Integrate ( int e^{sin x} cdot cos x d x )12
446Solve ( 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
449Integrate 🙁 int frac{d x}{x(x+1)} )12
450Evaluate 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
452If ( 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
454If ( 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
456Evaluate ( int frac{10 n^{9}+10^{n} ln 10}{sqrt{n^{10}+10^{n}+10^{10}}} d n )12
457Evaluate 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
459Evaluate
( 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
46113.
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
463If ( 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
46413. 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
465The 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
466a 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
468Evaluate the following integral as limit
of sums:
( int_{0}^{5}(x+1) d x )
12
469Evaluate: ( 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
471Integrate ( int frac{1}{x^{1 / 2}+x^{1 / 3}} d x )12
472Evaluate the given integral.
( int(x+1) e^{x} log left(x e^{x}right) d x )
12
473( int sec x d x )12
4748.
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
478Solve: ( int_{0}^{pi} sqrt{1-sin x} d x )12
479The value of ( left(int x cdot e^{-x} d xright) ) is,12
4802z dz
Evaluate:
32+1
12
481Solve:
( int frac{(x-1)}{(x+1)(x-2)} d x )
12
482Evaluate the following integral
( int frac{a}{b+c e^{x}} d x )
12
483The 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
484Let ( 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
486Show 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
487Integrate with respect to ( x ) ( int frac{1}{x^{6}left(1+x^{-5}right)^{frac{1}{5}}} d x )12
488The 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
489The 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
491Solve ( 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
492ff ( 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
493Integrate:
( 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
494If ( 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
495If ( 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
497Integrate ( int x log 2 x d x )12
498Solve: ( int frac{boldsymbol{d x}}{13+3 cos x+4 sin x} )12
499The 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
501The 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
503Evaluate ( int_{0}^{3} frac{x}{sqrt{x^{2}+16}} d x )12
504The 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
505Solve ( int x^{2} cos x d x )12
506Evaluate
( int frac{x^{3}+4 x^{2}-7 x+5}{x+2} d x )
12
507If 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
508If ( 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
509Evaluate 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
512Integrate the function ( frac{cos x}{sqrt{1+sin x}} )12
513If ( 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
514Evaluate ( int_{0}^{pi / 4} sec ^{2} x d x )12
515If ( 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
516Solve:( int frac{x}{sqrt{4-x^{2}}} d x )12
517Evaluate: ( int_{0}^{frac{pi}{2}} log left(frac{4+3 sin x}{4+3 cos x}right) d x )12
518Evaluate: ( int frac{x^{3}-x^{2}+x-1}{x-1} d x )12
519Find :
( int log x d x )
12
520Find ( int frac{sqrt{boldsymbol{x}}}{sqrt{boldsymbol{a}^{3}-boldsymbol{x}^{3}}} boldsymbol{d} boldsymbol{x} )12
521Solve ( : 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
524Evaluate ( int_{0}^{pi / 2} frac{cos ^{2} x}{1+3 sin ^{2} x} d x )12
525Evaluate: ( 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
527Assertion ( : 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
528The 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
529Integrate the rational function
( frac{x}{(x-1)^{2}(x+2)} )
12
53074
tan” x dx then lim n[In +In+2) equals
n->00
in +In+2] equals [2002]
(b)
1
(d) zero
(C)
12
53112.
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
532The 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
533Evaluate : ( int frac{d x}{a+b e^{c x}} )12
534Maximum value of ( g(x) ) in ( x in[0,7] ) is.
( A cdot 3 )
B. ( 9 / 2 )
( c .3 / 2 )
( D )
12
535The 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
537The value of ( left(int_{0}^{pi / 6} sec ^{2} x d xright)^{2} ) is:12
538Integrate the following function with
respect to ( x: )
( int frac{1}{left(x^{2}-1right)} d x )
12
539If ( 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
540Prove 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
542Integrate ( int x . sin 2 x d x )12
543Solve ( int_{pi / 2}^{3 pi / 2}[2 sin x] d x )12
544The 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
545Evaluate ( 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
547Integrate ( int_{a}^{b} e^{x} d x )12
548The 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
550The value of ( int_{0}^{pi} sin ^{50} x cos ^{49} x d x ) is
A .
в.
c. ( frac{pi}{2} )
D.
12
551Assertion ( (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
552Evaluate: ( int_{frac{pi}{2}}^{pi} frac{1-sin x}{1-cos x} d x )12
553If ( 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
554Integrate the following ( int frac{x^{2}}{(x sin x+cos x)^{2}} d x )12
555Solve:
( int frac{4 x+6}{2 x^{2}+5 x+3} d x )
12
5561.
101
J sin x | dx is
(a) 20 (6) 8
[2002
18
O
(c) 10
(d)
14
12
557Integrate ( 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
559Solve ( 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
564Find ( 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
566Solve ( 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
567sec²x
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
56822. Find det12
569The 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
570If ( 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
571The 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
573Using 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
575Evaluate ( 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
576Evaluate ( : int e^{sin ^{-1} x}left(frac{ln x}{sqrt{1-x^{2}}}+frac{1}{x}right) d x )12
577Evaluate 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
579Evaluate the following integrals:
( int sec ^{4} 2 x d x )
12
580The value of ( int_{-8}^{8}left(sin ^{93} x+x^{295}right) d x )
A .
B. –
c. 0
D.
12
581Find 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
583Integrate the rational function
( frac{boldsymbol{x}}{(boldsymbol{x}+mathbf{1})(boldsymbol{x}+mathbf{2})} )
12
584Integrate :-
( int log log x+frac{1}{(log x)^{2}} d x )
12
585Integrate the following w.r.t. ( x ) ( frac{1}{2 x+3} )12
586( int frac{1}{7 x+6} d x )12
587If ( 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
589The 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
590Evaluate ( int_{0}^{pi / 2} cos x d x )12
591Evaluate ( int frac{boldsymbol{d x}}{sqrt{mathbf{2 a x}-boldsymbol{x}^{2}}} )12
592The 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
593Evaluate the given integral: ( int_{0}^{1} x^{4} d x )12
594( int x^{9} d x )12
595Area 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
596ntegrate the function ( frac{mathbf{5} boldsymbol{x}+mathbf{3}}{sqrt{boldsymbol{x}^{2}+mathbf{4} boldsymbol{x}+mathbf{1 0}}} )12
597Solve:
( int sin ^{3} x cdot cos ^{2} x d x )
12
598Taking 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
600Antiderivative 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
601Evaluate:
( intleft(frac{x cos x+sin x}{x sin x}right) d x )
12
602Evaluate 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
603State 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
604Evaluate: ( int tan ^{-1} x d x )12
605Evaluate :
( int frac{x}{(x-1)^{2}(x+2)} d x )
12
606The 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
608Evaluate: ( int_{0}^{1} frac{1}{sqrt{1+x}-sqrt{x}} d x )12
609Integrate:
( int x sqrt{x^{2}+2} d x )
12
610Integrate:
( int_{0}^{pi} frac{d x}{5+3 cos x} )
12
611Find 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
613Integrate the following function:
( e^{x}left(frac{1+sin x}{1+cos x}right) )
12
614Let ( 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
615Evaluate: ( 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
616Evaluate:
( int x cos ^{3} x d x )
12
617Evaluate: ( int x^{-9} d x )12
618Evaluate ( 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
6191
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
620Let ( 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
621Solve:
( 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
623If ( 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
626Find ( : int frac{sin 2 x}{left(sin ^{2} x+1right)left(sin ^{2} x+3right)} d x )12
627Evaluate :
( 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
630Solve :
( int frac{cos x-sin x)}{(1+sin 2 x)} d x )
12
631Suppose 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
632If ( 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
633If ( 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
634x
+
– 2r
+
x
26.
Evalute
+ 1
dx. (1993 – 5 Marks
12
635Let ( 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
638Find ( 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
639The 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
640Solve :
( 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
642Evaluate 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
645Evaluate the definite integral ( int_{frac{pi}{6}}^{frac{pi}{3}} frac{sin x+cos x}{sqrt{sin 2 x}} d x )12
646The 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
647Integrate:
( 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
649Integrate:
( int sin ^{4} x d x )
12
650If ( 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
651Integrate: ( frac{3 x^{2}}{x^{6}+1} )12
652Integrate the function ( e^{x}(sin x+ )
( cos x )
12
6539.
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
654What 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
655The 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
657Evaluate 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
659The 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
661Let ( 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
662The 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
663By 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
664Write a value of
( int sqrt{x^{2}-9} d x )
12
665Evaluate ( intleft(frac{x^{2}+3 x+4}{sqrt{x}}right) d x )12
666Integrate the rational function ( frac{1}{left(e^{x}-1right)} )12
667Evaluate ( : 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
670Integrate the rational function
( frac{1}{xleft(x^{n}+1right)} )
12
671Evaluate:
( int sqrt{frac{boldsymbol{a}+boldsymbol{x}}{boldsymbol{a}-boldsymbol{x}}} boldsymbol{d} boldsymbol{x} )
12
672By 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
673Solve: ( 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
675Solve ( 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
678What is ( int_{-frac{pi}{2}}^{frac{pi}{2}}|sin x| d x ) equal to ( ? )
( A cdot 2 )
B. 1
( c . pi )
D.
12
679Evaluate ( int e^{x}left(frac{1+sin x}{1+cos x}right) )12
68015.
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
682Evaluate:
( int frac{x^{3}-4 x^{2}+6 x+5}{x^{2}-2 x+3} d x )
12
683Solve: ( 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
6857. S v1+ cos x dx equals
(a) 212 sin+C
(C) -2 12cos+c
(6) -212 sin+c
(d) 272.cos+c
12
686Evaluate 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
687Evaluate 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
689Integrate ( int frac{x^{3}-x^{2}+x-1}{x-1} d x )12
690ntegrate the function ( frac{1}{sqrt{8+3 x-x^{2}}} )12
691Solve :
( 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
694Solve ( 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
696What 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
697Integrate ( int frac{log x}{x^{2}} d x )12
698Evaluate ( : int_{-x}^{x}(cos a x-sin b x)^{2} d x )12
69927. 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
701Evaluate the following definite integral:
( int_{0}^{4}left(4 x-x^{2}right) d x )
12
702Evaluate ( int_{0}^{1} cot ^{-1}left(1-x+x^{2}right) d x )12
703Solve ( sqrt{frac{x^{2}-a^{2}}{x}} d x )12
704The 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
705Solve:
( 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
708Find ( int frac{d x}{xleft(x^{3}+1right)^{2}} )12
709Find 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
711A 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
713f ( 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
71538. 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
716Evaluate the integral ( int_{0}^{2}left(x^{2}+2 x+right. )
1) ( d x )
12
717Solve ( int(2 x+3)^{2} d x )12
718Evaluate the following definite integral:
( int_{-1}^{1} frac{1}{x^{2}+2 x+5} d x )
12
719Integrate ( 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
721Integrate ( int frac{3 x-1}{(x-1)(x-2)(x-3)} d x )12
722Solve :
( int frac{d x}{x^{2}+8 x+20} )
12
723Evaluate:
( 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
724Solve ( int_{0}^{pi / 2} frac{x sin x cos x}{cos ^{4} x+sin ^{4} x} d x )12
725Evaluate the following integral:
( int frac{x^{2}-1}{x^{2}+4} d x )
12
726Using (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
728If ( int frac{1}{5+4 cos 2 theta} d theta=A tan ^{-1}(B tan theta)+c )
then ( (A, B)= )
12
729The 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
732The 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
733dr
is equal to
cos x -sin x
(
tezlog col () +C
12
734Evaluate the following integral as limit
of sum:
( int_{0}^{5}(x+1) d x )
12
735If ( 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
736Evaluate: ( int frac{1}{x+sqrt{x}} d x )12
737Evaluate: ( int_{e}^{e^{2}} frac{d x}{x log x} )12
738If ( 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
740Evaluate ( int_{0}^{1}left(2 x^{2}+x+1right) d x )12
741Evaluate: ( int frac{5 x-2}{1+2 x+3 x} d x )12
742Evaluate: ( 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
743Evaluate 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
746If ( 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
748If ( 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
749Calculate the following integral
( int_{0}^{3}left[3^{1-x}+left(frac{1}{3}right)^{2 x-1}right] d x )
12
750State 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
752f ( 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
753If ( 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
755If ( 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
757Evaluate ( int_{0}^{2} x sqrt{x+2} d x )12
758If ( 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 ) equals12
760Integrate: ( int tan ^{3} x d x )12
761Integrate: ( int frac{x^{4}-x^{3}+8 x-8}{x^{2}-2 x+4} d x )12
762Using definite integration, find area of the triangle with vertices at ( A(1,1), B(3,3) A(1,1), B(3,3) )12
763Resolve ( 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
764Find 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
766Evaluate 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
76824. tesco is differentiable and [vas()dx =Şe, then s
(a) 2/5 (b) -5/2 @ 1 (d) 512->)
equals
12
769Solve:
( int_{0}^{frac{pi}{2}} x^{2} sin x d x )
12
770If ( 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
771Evaluate the given integral.
( int e^{x}(cot x+log sin x) d x )
12
772Evaluate :
( int sqrt{frac{1-cos 2 x}{1+cos 2 x}} d x )
12
773Solve: ( 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
775The 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
776Solve : ( int frac{1-cot x}{1+cot x} d x )12
777Solve: ( int_{0}^{pi / 2} frac{sin x d x}{(sin x+cos x)^{3}} )12
7786.
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, ) where12
780Find the antiderivative of the function
( left(sin frac{x}{2}+cos frac{x}{2}right)^{2} )
12
781If
( 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
78245. Evaluate Jelcosal (2 sin (= cos x) + 3 cos ( cosx)) sin x dx
(2005 2 Marks)
12
783Evaluate 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
784U
T4
TT/2
18.
Show that I f (sin 2x) sin x dx = V2 ) f(cos 2x) cos x de
(1990 – 4 Marks
12
785Evaluate ( 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
787The mean value of 6,9,12 is12
788Integrate 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
79032.
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
791Evaluate ( int frac{1}{sqrt{9-25 x^{2}}} d x )12
792Evaluate: ( int_{0}^{pi / 2} frac{sin ^{2} x}{sin x+cos x} d x )12
793The 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
794Solve: ( 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
796Evaluate the integral ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x )12
797The 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
798Solve :
( 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
800Evaluate: ( int_{0}^{1}left(8 x^{2}+16right) d x )12
801Integrate ( 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
803The 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
804Let ( 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
805Find
( int_{0}^{1 / 4 pi} ln (1+tan x) d x )
12
806Evaluate 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
809VINOJ
14.
For any natural number m, evaluate
|(x3m + x2m + x)(2x2m +3xm +6)/m dx ,x>0
12
810Evaluate: ( 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
811Evaluate; ( 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
815n-1
=1, 2, 3, ……. Then,
(2005
” (a) S. 57
| 6 ins
(6) Sa>
(0) T. 5
12
816Find ( 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
817Assertion
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
818Integrate the function ( x log 2 x )12
819Evaluate 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
820Find ( : int frac{sin ^{6} x}{cos ^{8} x} d x )12
8211. Integrate:12
822Solve: ( 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
8259. 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
826The 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
827If ( 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
828Solve ( int(2 t-4)^{-4} d t )12
829Evaluate ( int_{0}^{infty} sin x d x )12
830The 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
831Illustration 2.43 Evaluate V1+ y2 + 2y dy12
832Evaluate 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
833Evaluate ( 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
835Integrate 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
836Solve: ( intleft(boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}right)^{mathbf{3}} boldsymbol{d} boldsymbol{x} )12
837Evaluate ( 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
839OULD
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
840Evaluate 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
843What 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
8443. Evaluate [(elog x + sin x) cos x dx.
Evalua
dx
12
845Evaluate ( int frac{boldsymbol{d} boldsymbol{x}}{sqrt{mathbf{8}+mathbf{3} boldsymbol{x}-boldsymbol{x}^{2}}} )12
846Integrate:
( int frac{1+109 x}{x cdot 109 x} cdot d x )
12
847Evaluate the given integral. ( int e^{x} frac{x-4}{(x-2)^{2}} d x )12
848If ( 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
849If ( 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
850Evaluate ( int_{2}^{3} x^{2}+2 x+5 d x )12
851( int(3 x-2) sqrt{x^{2}+x+1} d x= )12
852The 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
856x²+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
858Evaluate: ( int_{3}^{9} frac{sqrt[3]{12-x}}{sqrt[3]{x}+sqrt[3]{12-x}} d x )12
859Evaluate: ( 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
860Statement-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
861Integrate the function ( frac{e^{5 log x}-e^{4 log x}}{e^{3 log x}-e^{2 log x}} )12
862Solve ( : int_{0}^{2} x sqrt{x+2} d x )12
863Evaluate ( int frac{cos x-sin x}{cos x+sin x} cdot(2+ )
( 2 sin 2 x) d x )
12
864Evaluate ( int_{0}^{2} frac{x}{3} d x )12
865If ( 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
866Using
(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
868et ( 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
869Find ‘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
871Integrate ( 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
873Evaluate the given integral: ( int_{0}^{5} x^{4} d x )12
874Find ( int sqrt{1+cos 2 x} d x )12
875Solve: ( 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
878The 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
881Evaluate 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
88312. The value of
12004
| 11-r2 dx is
14
12
884The value of ( int e^{ln sqrt{x}} d x ) is12
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
886Evaluate: ( 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
887Solve : ( int_{-2}^{2}|2 x+3| d x )12
888Evaluate the following integral:
( int_{-1}^{1}|2 x+1| d x )
12
889Evaluate ( 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
892Evaluate ( 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
895Evaluate 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
896eosx
37. Integrate
o e osx + e-cos x ax.
12
897Integrate the following function with
respect to ( x ) ( frac{sec ^{3} x}{csc x} )
12
898If ( 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
900Evaluate ( : int_{-2}^{2}|2 x+3| d x )12
901Evaluate ( 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
902The 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
904If ( 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
906Evaluate 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
90814. 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
909Solve ( : 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
911ITICS
14. If f() = {** sin x, for lets 2. then Š S(@dx =
otherwise,
-2
(a) I
(b) i
(c) 2
(2000)
(2) 3
12
912Evaluate:
( int_{-1}^{1} x e^{x^{2}} d x )
12
913Find :
( int frac{2}{(1-x)left(1+x^{2}right)} d x )
12
914Integrate ( int frac{d x}{x^{4}+1} )12
915Evaluate: ( int frac{boldsymbol{d t}}{left(1-t^{2}right)left(1-2 t^{2}right)} )12
916Solve ( 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
920Solve ( 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
922Integrate 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
925Evaluate the following definite integrals
( int_{0}^{1} frac{1}{1+x^{2}} d x )
12
926If ( 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
927Evaluate :
( int_{0}^{pi / 2} frac{sin x}{1+cos ^{2} x} d x )
12
928If ( 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
929Evaluate ( int frac{cos x}{(2+sin x)(3+4 sin x)} d x )12
930Solve ( 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
93325. 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
935Evaluate ( int_{0}^{2}left(3 x^{2}-2right) d x )12
936IF ( 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
937What 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
938Solve: ( int frac{d x}{left(2 x^{2}+3right)left(x^{2}-4right)} )12
939Evaluate 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
9402.
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
941Evaluate ( : quad I=int frac{x+9}{x^{2}+5} d x )12
942Evaluate: ( int_{0}^{2}left(x^{2}+3right) d x ) as limit of
sums
12
943Trs
23. Evaluate ) *sin 2x sin
coun) are o
Evaluate
– dx
a
2 x – 1
12
944d t is
(2010)
31. The value of lim li
1x³0x²
4+4
12
945Evaluate the following as the limit of
sum :
( int_{0}^{2}(x+4) d x )
A .4
B. 6
c. 8
D. 10
12
946evaluate :
[
boldsymbol{I}=int frac{2 x}{x^{2}-60 x+6} d x
]
12
947Solve 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
94916. Evaluate ſ log[V1- x + V1+x]dx12
950Evaluate the following definite integral:
( int_{1}^{2} e^{2 x}left(frac{1}{x}-frac{1}{2 x^{2}}right) d x )
12
951The domain of ( sin (cos theta) )
A. ( z )
в. ( R )
( c cdot Q )
D. ( N )
12
952Evaluate: ( intleft(frac{1}{(ln x)}-frac{1}{(ln x)^{2}}right) d x )
equals
12
953Solve:
( 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
954Let 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
956The 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
957Evaluate: ( int frac{1}{9 x^{2}+49} d x )12
958Value 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
959If ( 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
9608.
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
962Find ( int frac{d x}{xleft(x^{3}+1right)^{2}} )12
9636.
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
964Integrate
( int frac{x}{x^{2}+x+1} d x )
12
965( int frac{1}{(x+2)(x+3)} d x )12
966Integrate 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
968If ( 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
970Solve:
( 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
972Solve : ( 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
975Evaluate ( int_{0}^{pi} frac{x}{a^{2} cos ^{2} x+b^{2} sin ^{2} x} d x )12
976If ( 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 to12
978TU/sin x + cos x dx
6.
Evaluate : J 9+16 sin 2x
12
979Evaluate the following integrals:
( int_{0}^{pi} x d x )
12
980Solve ( 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
981Resolve ( 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
982If ( 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
983Find: ( int frac{4}{(x-2)left(x^{2}+4right)} d x )12
984Show 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
985X
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
986Solve: ( int_{0}^{1} frac{1}{2 x^{2}+x+1} d x )12
98722. 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
988The 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
9895. 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
99221. If y = 3×2 + 2x + 4, then ſy dx will be…12
99329. 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
995The question is refer to image.
8
12
996The 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
997If ( 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 to12
9995.
Evaluate the following | 204+1)3/4
12
1000( int_{0}^{1} frac{e^{x}}{1+e^{2 x}} d x )12
1001The 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
1003If ( 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
1004Evaluate ( int_{0}^{2}left(x^{2}-xright) d x )12
1005Ven3
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
1006Let ( 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
1007Integrate:
( int frac{1}{1+tan x} d x )
12
1008Evaluate 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
1009Evaluate 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
1013Evaluate the following integral:
( int frac{left(e^{sin ^{-1} x}right)^{2}}{sqrt{1-x^{2}}} d x )
12
1014Evaluate 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
1015Evaluate the following integral:
( int_{0}^{2}(3 x+2) d x )
12
1016The 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
1017Integrate the rational function
( frac{3 x-1}{(x-1)(x-2)(x-3)} )
12
1018If ( 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
1020State 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
102243. 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
1023Let ( 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
1026sin x
1.
Evaluate
dx
sin x – COS X
12
1027Evaluate: ( int frac{2 x}{(x+5)^{2}} d x )12
1028Find ( 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
1032Evaluate :
( 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
1034The 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
10372. Integrate the following: S(2t – 4)4 dt12
1038Find the value of ( int sec ^{2} x tan ^{3} x d x )12
1039Evaluate:
( int sec x tan x d x )
12
1040Prove 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
10412.
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
104332
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
1045Find 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
10471.
The value of the definite integral
‘) dx
is
(a)
(c)
– 1
1+e-1
(b) 2 (1981 – 2 Marks)
(d) none of these
12
1048Evaluate ( int_{0}^{2}|1-x| d x )12
1049Evaluate 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
1050Evaluate 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
1052Solve:
( 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
105442. 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
10551/2
13. Value of
cos 31 dt is
ما را به زیبا
12
1056If ( 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
1057Solve ( 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
1059The 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
1061Given 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
1063Evaluate ( int_{1}^{3}(3 x-2) d x )12
1064Evaluate : ( 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
1067Let ( 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
1068If ( 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 to12
1071If ( 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
1072Solve ( 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
1074When 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
107531/4
dx
To
(1999 – 2 Mark:
– is equal to
1+cos x
(6) 2
(a) 2
(c) 1/2
(d) -1/2
12
1076Evaluate ( 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
1078Evaluate ( 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
1079The 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
1080Write the formula for integration by
parts.
12
1081Evaluate the given integral: ( int_{0}^{1}(1- ) ( left.boldsymbol{x}^{2}right) boldsymbol{d} boldsymbol{x} )12
1082Assertion
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
1083Evaluate :
( intleft(1+2 x+3 x^{2}+4 x^{3}+dotsright) d x )
( |boldsymbol{x}|<mathbf{1}) )
12
108418. 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
1086Write 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
1091Solve :
( 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
1092Evaluate: ( int_{-1}^{2}left|x^{3}-xright| d x )12
1093Evaluate:
( int frac{sin ^{6} x+cos ^{6} x}{sin ^{2} x cos ^{2} x} d x )
12
1094Evaluate the following integration w.r.t.
( boldsymbol{x} )
( int frac{1}{(4 x+5)^{2}+1} d x )
12
1095Repeated 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
1096Solve
( 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
1099Obtain as the limit of sum ( int_{log _{e}^{3}} e^{x} d x )12
1100If ( 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
1101Evaluate ( : int frac{log x}{x} d x )12
1102Observe 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
1105Find ( 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
1109Evaluate ( 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
1112Integrate the function ( frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}} )12
1113Integrate:
( int frac{sin ^{-1} x}{sqrt{1-x^{2}}} d x )
12
1114The 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
1116If ( 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
1117Solve
( frac{1}{2} int frac{(-4+2 x)}{sqrt{5-4 x+x^{2}}} )
12
1118If ( 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
1119Evaluate ( 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
1120If ( 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
1122Evaluate ( int frac{d x}{sqrt{x+1}-sqrt{x}} )12
1123In (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
1125Evaluate ( 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
1127Suppose 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
1128Find: ( int x^{2} cdot log x d x )12
1129Evaluate ( int_{0}^{2}left(x^{2}+2 x+1right) d x )12
1130Evaluate ( : int_{-pi / 4}^{pi / 4} x^{5} cos ^{2} x d x )12
1131( int_{0}^{pi / 4} sec ^{2} x d x )12
1132The 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
1134Evaluate ( : 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
1136ntegrate the function ( frac{1}{sqrt{9 x^{2}+6 x+5}} )12
113711. If y = x², then
yox will be:
(d) O
12
1138Find 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
1139Evaluate ( 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
114013a + 4x²
44. Find the value of J
-1/3 2-
dx
12
1141Let ( 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
1142Evaluate the following integral:
( int frac{x^{3}-3 x^{2}+5 x-7+x^{2} a^{x}}{2 x^{2}} d x )
12
114335. 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
1148The 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
1149Integrate:
( 2 x^{2} e^{x^{2}} )
12
1150Evaluate 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
1152Solve :
( int x sqrt{1+2 x^{2}} d x )
12
1153Evaluate ( : int frac{e^{2 x}-1}{e^{2 x}+1} d x )12
11541 (x-1) et
4.
Evaluate :
(x+13 ax
12
1155Solve ( int_{1}^{-1} frac{d}{d x} tan ^{-1}left(frac{1}{x}right) d x )12
1156Evaluate: ( int frac{sec ^{8} x}{operatorname{cosec} x} d x )12
1157Evaluate ( int_{0}^{pi / 2} frac{cos x}{1+sin ^{2} x} d x )12
1158Write 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
116020. 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
1161If ( 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
1162The 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
1163Prove that:
( int frac{x^{2} d x}{(x sin x+cos x)^{2}} )
12
1164Evaluate ( int(1-x) sqrt{x} . d x )12
1165Let 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
1166Prove that:
( int_{0}^{pi} frac{x d x}{1+sin x}=pi )
12
1167Solve:
( 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
1169Solve ( 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
1171If ( 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
1172If ( 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
1173If /(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
1175Solve:
( 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
1177Solve:
( int frac{d x}{sqrt{4-x^{2}}} )
12
1178The 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
1179Solve:
( int frac{cos x}{6+4 sin x-cos ^{2} x} d x )
12
1180If ( 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
1182If ( 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
1183Solve:
( int frac{x^{2}+1}{x^{2}-4 x+6} d x )
12
1184Consider 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
1185Solve ( 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
1187Evaluate 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
1190Solve ( 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
1193Evaluate the following definite integral:
( int_{0}^{2} frac{1}{4+x-x^{2}} d x )
12
1194Evaluate the following integral as limit
of sum:
( int_{1}^{4}left(x^{2}-xright) d x )
12
1195Evaluate ( int frac{d x}{x+4-x^{2}} )12
1196If ( 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
1197Evaluate 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
1198c
os
+ sin
10. Find the indefinite integral ſcos 20 ln
cos 0 – sino)
indefinite integral co
12
1199Evaluate the following integral
( int frac{e^{x}+1}{e^{x}+x} d x )
12
1200Integrate ( int_{1}^{2} x^{2} d x )12
1201Evaluate ( : int frac{1+log x}{x(2+log x)(3+log x)} d x )12
1202Evaluate: ( int frac{d x}{left(x^{2}+1right)left(x^{2}+4right)} )12
1203Evaluate the given integral. ( int frac{x^{2}-1}{x^{4}+x^{2}+1} d x )12
1204Evaluate the integral ( int_{1}^{2}left(frac{1}{x}-frac{1}{2 x^{2}}right) e^{2 x} d x ) using
substitution.
12
1205Evaluate the following integral:
( int_{0}^{3} x^{2} d x )
12
1206I : 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
1208Evaluate ( int_{0}^{2 / 3} frac{d x}{left(4+9 x^{2}right)} )12
1209Evaluate :
( 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
1211The 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
1213Prove that ( int_{0}^{pi / 2} frac{d x}{1+tan x}=frac{pi}{4} )12
1214Solve ( 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
1215Prove 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
1216Find ( int frac{x^{4}+1}{xleft(x^{2}+1right)^{2}} )12
1217Suppose 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
1218Let ( 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
1219If ( 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
1220Evaluate 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
1221If ( 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
1222Evaluate the following functions w.r.t. ( intleft(3 x^{2}-5right)^{2} d x )12
1223Evaluate ( 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
1226Evaluate: ( int frac{d x}{1-tan x} )12
1227Evaluate ( 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
1230Find 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
1232Evaluate:
( int[sin (log x)+cos (log x)] d x )
12
1233The 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
1235Evaluate: ( 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
1237Solve ( 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
1239If 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
1240Evaluate 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
124110. 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
1243Evaluate
( 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
1244integrate:
[
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
1247Evaluate ( : int frac{x-3}{(x-1)^{3}} e^{x} d x )12
1248Evaluate: ( int frac{cos ^{3} x}{sin ^{2} x+sin x} )12
1249r/2
dx
7.
(1993 – 1 Marks)
is
The value of
(a) o
Ö
1+tan
(6) 1
(c) r12
(d) a 14
12
1250If ( 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
1251If ( 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
1252Find ( : int frac{(2 x-5) e^{2 x}}{(2 x-3)^{3}} d x )12
1253Evaluate: ( int_{0}^{2} frac{e^{x}}{e^{2 x}+1} d x )12
1254Evaluate ( : int frac{sec x}{1+operatorname{cosec} x} d x )12
1255Solve ( int frac{x}{sqrt{x+4}} d x )12
1256Evaluate: ( int frac{2 x}{left(x^{2}+4right)} d x )12
1257Integrate ( int frac{1}{sqrt{3-x^{2}}} d x )12
1258Evaluate the following integrals.
( int frac{d x}{sqrt{2 x-3 x^{2}+1}} )
12
1259Evaluate 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
1260If ( 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
1262Evaluate ( int e^{x}(tan x-log cos x) d x )12
1263Evaluate ( 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
126627. The solution for x of the equation
“INOP-1 3 is (2007)
at is [2007]
(a) v3 (b) 272 ( 2 () None
12
1267What 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
1269Evaluate the integral ( int_{1}^{2} sqrt{(x-1)(2-x)} d x )
A. ( frac{pi}{8} )
в.
c. ( frac{1}{8} )
D.
12
1270Differentiate 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
1271The 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
1272Find 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
1273Evaluate :
( 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
1275If ( 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
1276Integral of ( frac{left(4 x^{2}-2 sqrt{x}right)}{x}+frac{1}{1+x^{2}}- )
5 ( operatorname{cosec}^{2} x ) is
12
1277The 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
1279Simplify:( intleft(frac{1}{(ln x)}-frac{1}{(ln x)^{2}}right) d x )12
1280Evaluate ( 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
1281Let ( 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
1282Evaluate:
( 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
1284Solve:( 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
1287Integrate 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
12895.
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
1290Evaluate: ( 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
1291If ( 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
1293Solve: ( 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
1294If ( 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
1295Solve: ( 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
1300Number 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
1302Evaluate:
( int e^{x^{3}+x^{2}-1}left(3 x^{4}+2 x^{3}+2 xright) d x )
12
1303Evaluate
( int frac{x+1}{x^{2}+3 x+12} d x )
12
1304Show that: ( int_{0}^{frac{pi}{4}} log (1+tan x) d x= )
( frac{pi}{8} log 2 )
12
1305Evaluate
( int sin ^{-1} frac{2 x}{1+x^{2}} d x )
12
1306Solve :
( int tan ^{2}(2 x-3) d x )
12
1307Integrate 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
1311Find:
( 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
1314Solve: ( int cos ^{3} x d x )12
1315If ( 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
1317Resolve ( 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
1319Evaluate: ( 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
1321Integrate 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
1324Evaluate ( int r^{4}left(7-frac{r^{5}}{10}right) d r )12
1325Match the integrals of ( f(x) ) if12
1326Evaluate the given integral. ( int x^{2} cos x d x )12
1327Integrate ( 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
1330Evaluate: ( int frac{x e^{x}}{(x+1)^{2}} d x )12
1331Evaluate 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
1334Evaluate 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
13355.
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
1336Integrate the function ( sqrt{1-4 x-x^{2}} )12
1337Find 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
1338Find 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
1340If ( 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
1341Evaluate 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
1343Solve:
( int frac{1}{x} sqrt{frac{x-1}{x+1}} d x )
12
1344Solve :
( int frac{x}{2 x-3} d x )
12
1345Evaluate ( 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
1347Solve
( 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
1350Evaluate: ( 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
1351Evaluate: ( 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
1352Solve ( 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
1354The 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
1355ſa-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
1357If ( 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
1358Solve:
( int frac{sqrt{x}-sqrt{a}}{sqrt{x+a}} d x )
12
1359Find 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
1361Assertion
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
136450. 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
1366pr 2x(1+sin x)
33. Determine the value of |”.
J-TT
1+cOS X.
12
1367Solve ( int frac{3 x-1}{(x+2)^{2}} d x )12
1368The 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
1370Solve ( 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
1372Integrate ( int_{0}^{1}left(3 x^{2}+2 xright) d x )12
1373Integrate 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
1375Evaluate 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
1376The 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
1377Solve :
( int frac{x^{2}+3 x-1}{(x+1)^{2}} d x )
12
13781/2
14. If | xf (sin x)dx = A ( f (sin x)dx, then A is
12004
(a) 20
(6)
(d) 0
12
1379Evaluate ( : 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
1380The 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
1381Integrate ( int_{0}^{2}left(x^{2}+xright) d x )12
1382If ( 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
1383If 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
1384By using the properties of definite integrals, evaluate the integral ( int_{0}^{frac{pi}{2}} cos ^{2} x d x )12
1385Solve ( left[-int_{0}^{pi / 2} cos left(frac{pi}{4}+frac{x}{2}right) e^{x}right] d x )12
1386Evaluate the following definite integral:
( int_{pi / 6}^{pi / 4} operatorname{cosec} x d x )
12
1387Evaluate ( int_{0}^{5} x^{4} d x )12
1388Solve ( : 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
1392Evaluate the following integral:
( int sec ^{4} x tan x d x )
12
13936.
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
1394Suppose ( 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
1395Solve :
( boldsymbol{I}=int frac{boldsymbol{x}+mathbf{9}}{boldsymbol{x}^{2}+mathbf{5}} boldsymbol{d} boldsymbol{x} )
12
1396Solve ( 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
1397The 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
1398Evaluate :
( int frac{sec ^{2} x}{tan x} d x )
12
13991U 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
1400If 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
1401ntegrate the function ( frac{mathbf{5 x}}{(x+1)left(x^{2}+9right)} )12
1402The 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
1403If ( 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
1404Evaluate: ( int tan ^{-1} sqrt{x} d x )12
1405If ( 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
1406Evaluate ( : 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
1408Solve ( 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
1410Integrate: ( left(frac{a}{sqrt{x}}+2 b sqrt[3]{x^{2}}right) ) w.r.t12
1411Find the integral of
( intleft(sqrt{boldsymbol{x}}-frac{mathbf{1}}{sqrt{boldsymbol{x}}}right)^{2} boldsymbol{d} boldsymbol{x} )
12
1412Evaluate:
( int x+5 d x )
12
1413( int x^{3} tan ^{-1} x d x )12
1414Integrate ( sec ^{3} x ) w.r.t. ( x )12
141514. 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
1417Evaluate
( int frac{x^{3}}{sqrt{1+2 x^{4}}} d x )
12
1418If ( 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
1419Evaluate the following: ( int(1+x) e^{x} d x )12
1420If ( 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
1421The 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
1422Solve: ( int frac{x+sqrt{1-x^{2}}}{x sqrt{1-x^{2}}} d x=? )12
1423Evaluate ( : 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
1425Evaluate ( int frac{left(1+2 x^{2}right) d x}{x^{2}left(1+x^{2}right)} )12
1426Solve: ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{boldsymbol{x}+mathbf{1}}+sqrt{boldsymbol{x}}} )12
1427Evaluate ( int frac{x^{3}+1}{x^{2}+1} d x= )12
1428Integrate ( int e^{log (sec x+tan x)} sqrt{1+tan ^{2} x} d x )12
1429Solve :
( 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
1430Solve ( 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
1431If ( 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
1432Integrate the function ( tan ^{-1} sqrt{frac{1-x}{1+x}} )12
1433sec?
f(t)dt
– equals
31.
lim
-2
(2007 – 3 marks)
se o
re
3,7 ca are
12
1434Evaluate ( int_{1}^{2}left(x^{2}-1right) d x )12
1435The 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
1436Solve the equation:( int_{0}^{2}left[x^{2}-x+1right] d x )12
1437If ( 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
1438Solve ( int frac{x^{2}+5 x-1}{sqrt{x}} d x )12
1439Evaluate: ( int frac{2 x^{2}+1}{x^{2}left(x^{2}+4right)} d x )12
1440The 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
1441Evaluate 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
1443If ( 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
1444Evaluate ( 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
1446The 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
1447If ( 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
1449A 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
1450Evaluate 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
1452The 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
1454Solve:
( 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
145517. 117 – 12°dt, = {2de, 15 = 2* dr and
1a = \$ 2 1
(b) 11 > 12 (©)
13 = 14
(d) 13>I,
12
1456Solve: ( 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
1459If ( 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
1460Evaluate ( int_{-1}^{1} log frac{2-x}{2+x} d x )12
1461Find ( int a^{x} cdot e^{x} d x )12
1462Solve ( int_{0}^{1}|boldsymbol{x}| boldsymbol{d} boldsymbol{x} )12
1463( int e^{a x} cdot sin (b x+c) d x )12
1464If ( 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
1465If ( 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
1466Find: ( int_{-pi}^{pi} frac{cos ^{2} x d x}{1+a^{x}} ) where ( a>0 )12
1467Solve: ( int frac{cos x}{x} d x )12
146844. 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
146910.
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
14711/2
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
1472If ( 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
1473Integrate the function ( frac{x+2}{sqrt{x^{2}-1}} )12
1474Evaluate ( 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
1475Integrate 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
147812.
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
1479Evaluate: ( 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
1481If ( 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
1482Evaluate: ( int frac{x+3}{(x-1)left(x^{2}+1right)} d x )12
1483Integrate ( frac{(x+12)}{(x+1)^{2}(x-2)} )12
1484Solve:
( boldsymbol{I}=int frac{boldsymbol{d} boldsymbol{x}}{(boldsymbol{x}+mathbf{1}) sqrt{mathbf{1}-boldsymbol{x}^{2}}} )
12
1485Integrate ( 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
1489Evaluate ( 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
14904.
3/2
Find the value of 1 |x sin at x | dx
12
1491Set 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
1492Evaluate:
( int_{-1}^{1} sin ^{5} x cos ^{4} x d x )
12
1493Evaluate 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
1494Solve:
( int sqrt{(x-2)(x-2)} d x )
12
1495Evaluate:
( 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
1497Evaluate the definite integral ( int_{0}^{1} frac{boldsymbol{d} boldsymbol{x}}{sqrt{mathbf{1}+boldsymbol{x}}-sqrt{boldsymbol{x}}} )12
1498If 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
1499If ( 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
1500Evaluate ( int_{0}^{1} frac{1}{sqrt{1+x}-sqrt{x}} d x )12
1501If ( 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
1503The 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
1504Integrate ( 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
1505Solve ( 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
1508Prove 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
1511Evaluate:
( int x log (x+1) d x )
12
1512Obtain: ( int frac{(3 x+2)}{(x+1)(x+2)(x-3)} d x )12
1513If ( 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
1514ff ( 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
151512
(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
1517A 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
1518Evaluate the given definite integrals as
limit of sums:
( int_{0}^{4}left(x+e^{2 x}right) d x )
12
1519Find ( int frac{x^{4}}{(x-1)left(x^{2}+1right)} d x )12
1520Solve :
( int frac{u}{v} d x )
12
1521Evaluate ( 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
1523Evaluate: ( 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
1525Solve ( 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
1528Evaluate : ( 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
1529If ( 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
1531If ( 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
1532ff ( 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
1533Suppose ( 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
1534If ( 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
1535Prove that ( : int frac{1}{a^{2}-x^{2}} d x= )
( frac{1}{2 a} ln left|frac{a+x}{a-x}right|+c )
12
1536Assertion
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
1537Find: ( int frac{left(x^{4}-xright)^{frac{1}{4}}}{x^{5}} d x )12
1538Evaluate 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
1539The 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
1540Evaluate ( int_{0}^{1} e^{2-3 x} d x ) as a limit of a
sum
12
1541Evaluate the following integral:
( int_{0}^{2}left(x^{2}+3right) d x )
12
1542sinx
49. The value of
Bica
undx is :
(JEEM
1+
(b) 47
(C)
12
1543Evaluate the given integral. ( int sin ^{-1}left(frac{2 x}{1+x^{2}}right) d x )12
1544Evaluate: ( 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
1547Evaluate ( int_{0}^{2}left(x^{2}-3 x+2right) d x )12
1548Solve ( 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
1550Evaluate 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
1551Solve ( :left(x^{2}-1right) frac{d y}{d x}+2 x y=frac{1}{x^{2}-1} )12
1552The 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
1553Evaluate ( int_{4}^{12} x d x )12
1554Find an anti derivative (or integral) of the given function by the method of inspection.
( e^{2 x} )
12
1555If ( 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
1556Evaluate the definite integral ( int_{0}^{frac{pi}{4}} tan x d x )12
1557The 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
1558Evaluate:
( int_{0}^{1} x^{3}+3 x^{2} d x )
12
1559Evaluate : ( 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
1560Simplify:( int x ell n sqrt{x} d x )12
1561If ( 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
156219.
(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
1563Evaluate 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
1565Evaluate ( int_{-pi}^{pi} frac{2 x(1+sin x)}{1+cos ^{2} x} d x )12
15665.
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
1567nit+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
1569Number 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
1570Prove ( 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
1572The 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
1575ntegrate the function ( frac{mathbf{5 x}}{(x+1)left(x^{2}+9right)} )12
1576Solve it
( mathbf{2} boldsymbol{I}=int_{boldsymbol{O}}^{boldsymbol{Q}} boldsymbol{d} boldsymbol{x} )
12
1577Evaluate: ( int frac{tan x}{(cos x)^{2}} d x )12
1578If ( 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
1579The 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
1580Evaluate:
( int frac{x-x^{2}}{x^{2}-2 x-3} d x )
12
1581210
The value of
[2 Sinx) dx where [.) represents the greatest
integer function is
(1995)
(d) -2
(a) -ST
(6) – T
(
54
12
1582The 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
1584If ( 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
1585umbers), 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
1586If ( 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
1587Evaluate ( int frac{x^{3}+4 x^{2}-7 x+5}{x+2} d x )12
1588Match the column12
1589Evaluate: ( 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
1590Evaluate ( 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
1591Evaluate ( int x cdot e^{-x^{2}} d x )12
1592Evaluate ( 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
1594Find ( intleft(sqrt{boldsymbol{x}}+frac{mathbf{1}}{sqrt{boldsymbol{x}}}right) boldsymbol{d} boldsymbol{x} )12
1595Find 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
1596Integrate with respect to ( x ).
( e^{x} sin x )
12
1597Evaluate ( : 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
1599The 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
1600Evalaute 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
1601Let ( 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
1602Assertion
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
1603T/2
36. The value of the integral
Ttx
x² + In
22
cos xdx is
TT – X
(2012)
2
+4
12
1604f ( 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
1605Assertion
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
1606Solve:
( 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
1607Evaluate :
( 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
1611f ( int_{1}^{a}left(3 x^{2}+2 x+1right) d x=11, ) find real
values of a.
12
1612Evaluate the definite integral:
( int_{0}^{pi / 2} cos ^{2} x d x )
12
1613Solve ( int frac{x^{2} tan ^{-1} x^{3}}{1+x^{6}} d x )12
1614( int frac{d x}{16 x^{2}-25} )12
1615If ( 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
1617The 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
1618Find 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
1620Prove 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
1622Evaluate: ( 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
1624If ( 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
1625Evaluate the integral ( int_{-1}^{1} frac{d x}{x^{2}+2 x+5} ) using substitution12
1626Using 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
1627Consider 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
1628Find the integral ( int frac{2 x}{xleft(x^{2}+1right)} d x )12
1629If ( 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
1630Find area of ( boldsymbol{y}=boldsymbol{x}^{2} ) from ( boldsymbol{x}=boldsymbol{2} ) to ( boldsymbol{x}=boldsymbol{4} )12
1631Evaluate ( int frac{x cos ^{-1} x}{sqrt{1-x^{2}}} d x )12
1632Evaluate ( 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
1635Evaluate ( : intleft(x+frac{1}{x}right)^{3} d x, x>0 )12
1636Solve ( : int_{0}^{frac{pi}{2}} tan ^{5} x cos ^{8} x d x )12
1637Evaluate ( int frac{d x}{sqrt{5 x^{2}-2 x}} )12
1638Which 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
1639The 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
1640Evaluate 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
1642Evaluate: ( 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
1644Evaluate: ( int sqrt{frac{boldsymbol{x}-mathbf{5}}{boldsymbol{x}-mathbf{9}}} boldsymbol{d} boldsymbol{x} )12
1645Solve ( int frac{cos ^{2} theta d theta}{cos ^{2} theta+4 sin ^{2} theta} )12
164619. 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
1647The value of ( int frac{d x}{sin x cdot sin (x+alpha)} ) is equal
to
12
1648The 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
164940. 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
1653Evaluate the given integral. ( int e^{x}left(frac{sin x cos x-1}{sin ^{2} x}right) d x )12
1654If 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
1655I 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
1657Evaluate the following as the limit of
sum :
( int_{0}^{2}(x+4) d x )
A .4
B. 6
c. 8
D. 10
12
1658Evaluate :
( 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
1660Find ( int sqrt{boldsymbol{x}}left(boldsymbol{3} boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{3}right) boldsymbol{d} boldsymbol{x} )12
166116. Evaluate (log[V1- x + V1+x]dx12
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
1663Assertion 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
1664Evaluate ( int_{1}^{4}(x-1) d x )12
1665ntegrate the function ( frac{sec ^{2} x}{sqrt{tan ^{2} x+4}} )12
1666Evaluate the following integral
( int frac{sec x operatorname{cosec} x}{log (tan x)} d x )
12
1667Evaluate the integral:
( int_{0}^{2} frac{x}{3} d x )
12
1668Solve 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
1670Evaluate 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
1672The 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
1674Evaluate ( int frac{d x}{x^{2}-4 x+13} )12
1675( int cos ^{2} theta d theta=? )12
1676Evaluate 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
1679Evaluate 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
1684Show that ( int_{0}^{1} frac{log x}{sqrt{left(1-x^{2}right)}} d x=frac{pi}{2} cdot log frac{1}{2} )12
1685The 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
168634. 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
1687Evaluate
( int frac{x+2}{sqrt{x^{2}+4 x+1}} cdot d x )
12
1688Evaluate
( int frac{d x}{left(x^{2}+1right) sqrt{x^{2}+1}} )
12
1689If ( 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
1690Evaluate 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
1691V2 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
169220.5(_3 +2x )de is equal to..12
1693Evaluate ( int frac{2 cos x-3 sin x}{4 cos x+5 sin x} d x )12
1694Integrate:
( int sin x^{2} d x )
12
1695Evaluate : ( 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
1697If ( 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
1698Evaluate 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
1700Evaluate ( 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
1702The 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
1703The 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
1705Solve:
( 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
170941. 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
1710Find 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
1711Integrate:
( int_{0}^{pi} frac{d x}{5+3 cos x} )
12
1712If ( 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
1713Let ( 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
1714Integrate ( 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
1717If ( 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
1718The 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
1719Evaluate the following integral
( int frac{operatorname{cosec} x}{log tan frac{x}{2}} d x )
12
1720Integrate ( int frac{d x}{(x+1)(x+5)} )12
1721The 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
1722If 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
1724Evaluate ( int_{-1}^{1} 5 x^{4} sqrt{x^{5}+1} d x )12
1725Integrate: ( int frac{x}{x^{4}-x^{2}+1} d x )12
1726If ( 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
1727Evaluate ( 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
1728Evaluate ( intleft(frac{x^{6}-1}{1+x^{2}}right) d x ) for ( x in R )12
1729Solve ( int frac{2 x ln left(x^{2}-1right)}{left(x^{2}-1right)} d x )12
1730The 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
1733Evaluate the following definite integral:
( int_{0}^{pi / 2} sin x cos x d x )
12
1734Evaluate ( int_{0}^{1} x^{4} d x )12
1735Prove that
( int frac{x^{2}}{x^{6}+1} d x )
12
1736Assertion
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
1738If ( 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
1739Evaluate ( : int x tan ^{-1} x d x )12
1740Evaluate: ( int_{0}^{pi / 2} cos ^{2} x d x )12
1741Evaluate 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
1743If ( y=2^{2} 3^{2 x} 5^{-5} 7^{-5} ) then ( frac{d y}{d x}= )12
1744Integrate the function ( x sin 3 x )12
1745If ( 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
1747Solve ( 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
1748f
(x+1)
11. Evaluate x01 + xet 2 dx.
12
174918. 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
1750Prove 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
1752Integrate ( 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
1755Evaluate : ( int frac{(3 x+5) d x}{sqrt{x^{2}+4 x+3}} )12
1756Evaluate ( int frac{d x}{9 x^{2}+6 x+5} )12
1757Integrate:
( 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
175832.
[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
1762Evaluate: ( 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
1764Solve
( intleft(frac{x-1}{x+1}right)^{4} d x= )
12
1765If ( 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
1766Evaluate the following definite integral:
( int_{0}^{1} x+x^{2} d x )
12
1767For 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
1768Evaluate ( int frac{sin theta}{sin 3 theta} d theta )12
1769Evaluate:
( int_{0}^{2}left[x^{2}right] d x )
12
1770Evaluate:
( 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
1771Solve ( 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
1773Evaluate: ( 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
1774TO
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
1775Integrate the function ( frac{x+2}{sqrt{4 x-x^{2}}} )12
1776Find the value of ( int_{0}^{frac{pi}{2}} log (tan x) d x )12
1777Solve ( int_{b}^{a} frac{x}{sqrt{a^{2}+x^{2}}} d x )12
17781.
If S* (t)dt=x+S + f(t) dt, then the value of f(1) is
(1998 – 2 Man
12
1779Evaluate: ( int_{0}^{pi / 4} log (1+tan theta) d theta )12
1780Evaluate ( int_{0}^{pi / 2} cos x d x )12
1781Evaluate: ( intleft(3 x^{2}-5right)^{2} d x )12
178214. (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
1784Solve: ( int_{0}^{1} frac{1}{2 x^{2}+x+1} d x )12
1785solve :
( int(a x+b)^{2} d x )
12
1786Evaluate
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
1788The 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
1789Evaluate: ( int e^{x}(tan x+log (sec x)) d x )12
1790Integrate:
( frac{e^{2 x}-1}{e^{2 x}+1} )
12
1791Integrate the function ( sqrt{x^{2}+4 x-5} )12
1792Integrate ( : int frac{1+tan x}{x+log sec x} d x )12
1793Find 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
1796Evaluate 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
1799If ( 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
1801What 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
1803If ( 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
1805If ( 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
1806The 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
1807Integrate ( : int x sin ^{2} x )12
1808Integrate the function ( frac{x cos ^{-1} x}{sqrt{1-x^{2}}} )12
1809Assertion
( 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
1810Evaluate
( int sqrt{1+t^{3}} d t )
12
181119. Jx3 dx is equal to…12
1812Solve ( 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
1814Evaluate:
( 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
1816Solve ( 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
1819Assertion
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
1820Find ( int frac{sin ^{6} x}{cos ^{8} x} d x )12
1821Solve: ( int frac{sqrt{1+x^{2}}}{x^{4}} d x )12
1822Evaluate ( int_{1}^{2} frac{-1}{x^{2}} d x )12
1823The 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
1824Evaluate: ( int_{2}^{3} frac{1}{x} d x )12
1825If ( 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
1827Evaluate ( : int frac{d x}{1+cos a cos x} )12
1828Write 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
1829Illustration 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
1830Write a value of
( int sqrt{9+x^{2}} d x )
12
1831If ( 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
1832If ( 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
1833Evaluate ( int_{0}^{pi / 4} sin ^{3} 2 t cos 2 t d t )12
1834The 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
1836Evaluate ( 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
1838Evaluate: ( int_{-a}^{a} frac{sqrt{a-x}}{sqrt{a+x}} d x )12
1839The 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
1840Find 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
1841The 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
1842Evaluate:
( int frac{1}{a^{x} b^{x}} d x )
12
1843Evaluate ( int_{0}^{1}left(2 x^{2}+x+1right) d x )12
1844Evaluate 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
1846Assertion
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
1847Evaluate ( 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
1848Find the integral of ( intleft(2 x^{2}-3 sin x+right. ) ( mathbf{5} sqrt{boldsymbol{x}}) boldsymbol{d} boldsymbol{x} )12
1849Evaluate:
( int_{2}^{1}|boldsymbol{x}-mathbf{3}| boldsymbol{d} boldsymbol{x} )
12
1850A 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
1852Evaluate ( int frac{1}{sqrt{3} sin x+cos x} d x )12
1853Evaluate the following definite integral:
( int_{0}^{4} 4 x-x^{2} d x )
12
18542.
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
1855If ( 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
1856Evaluate the integral ( int_{0}^{1}left(1-x^{2}right) d x )12
1857Solve ( : int sec ^{-1} sqrt{x} d x )12
1858Find 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
1859Find 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
186013. Evaluate sin
1
2x+2
2x+2
dx
4x² + 8x+13)
12
1861Evaluate the integral ( int_{3}^{5}(2-x) d x )12
1862Find ( intleft(x^{2}+1right) d x )12
1863Evaluate 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
1864Evaluate ( 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
1866Solve ( int frac{1}{x^{5}}left(1+x^{4}right) d x )12
1867Solve:
( 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
1869If ( 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
1870The 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
1873If ( 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
1874Evaluate ( int frac{cos ^{2} x}{sin ^{3} xleft(sin ^{5} x+cos ^{5} xright)^{frac{3}{5}}} d x )12
1875ntegrate 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
1877Write a value of ( int e^{x}left(frac{1}{x}-frac{1}{x^{2}}right) d x )12
1878Evaluate: ( int_{0}^{frac{pi}{2}} frac{sin x-cos x}{1+sin x cos x} d x )12
1879Solve:
( int frac{log x}{x^{2}} d x )
12
1880Assertion
( 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
1881The 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
1882Evaluate ( int_{a}^{b} x sin x d x )12
1883( int_{0}^{frac{2}{3}} frac{d x}{4+9 x^{2}}= )12
1884If ( 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
1888If
( 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
1890Number of real solution of the given equation for ( x, int x^{2} e^{x} d x=0 )12
1891Evaluate 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
1893Evaluate ( 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
1895If ( 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
1898What 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
1899Find the partial fraction ( frac{2 x+1}{(3 x+2)left(4 x^{2}+5 x+6right)} )12
1900Evaluate: ( 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
1902show jus cinse) dr = Frosine) ds.
(1982 – 2 Mai
12
1903The 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
1905If ( 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
1906Evaluate the given integral.
( int frac{2}{1-cos 2 x} d x )
12
190730. Evaluate the definite integral :
– 1/3 (1-x²
12
1908Integrate the rational function ( frac{x^{3}+x+1}{x^{2}-1} )12
1909The 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
1910If ( 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
1911Find ( : int log x cdot d x )12
1912Evaluate the given integral. ( int x sin 2 x d x )12
1913Evaluate 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
1915For 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
1917Let ( 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
1918If ( 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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