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Indefinite Integrals

111 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Evening Slot
If $\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( \theta \right)} \right| + C$

where C is a constant of integration, then the ordered pair ($\lambda $, ƒ($\theta $)) is equal to :
A.
(–1, 1 – tan$\theta $)
B.
(1, 1 + tan$\theta $)
C.
(–1, 1 + tan$\theta $)
D.
(1, 1 – tan$\theta $)
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
The integral $\int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}} $ is equal to :
(where C is a constant of integration)
A.
${1 \over 2}{\left( {{{x - 3} \over {x + 4}}} \right)^{{3 \over 7}}} + C$
B.
${\left( {{{x - 3} \over {x + 4}}} \right)^{{1 \over 7}}} + C$
C.
$ - {1 \over {13}}{\left( {{{x - 3} \over {x + 4}}} \right)^{{{13} \over 7}}} + C$
D.
-${\left( {{{x - 3} \over {x + 4}}} \right)^{-{1 \over 7}}} + C$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 9th January Morning Slot
If ƒ'(x) = tan–1(secx + tanx), $ - {\pi \over 2} < x < {\pi \over 2}$,
and ƒ(0) = 0, then ƒ(1) is equal to :
A.
${1 \over 4}$
B.
${{\pi - 1} \over 4}$
C.
${{\pi + 1} \over 4}$
D.
${{\pi + 2} \over 4}$
2020 JEE Mains MCQ
JEE Main 2020 (Online) 8th January Morning Slot
If $\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c$

where c is a constant of integration, then $\lambda f\left( {{\pi \over 3}} \right)$ is equal to
A.
${9 \over 8}$
B.
2
C.
-2
D.
$-{9 \over 8}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Evening Slot
Let $a \in \left( {0,{\pi \over 2}} \right)$ be fixed. If the integral

$\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx$ = A(x) cos 2$\alpha $ + B(x) sin 2$\alpha $ + C, where C is a

constant of integration, then the functions A(x) and B(x) are respectively :
A.
$x - \alpha $ and ${\log _e}\left| {\cos \left( {x - \alpha } \right)} \right|$
B.
$x + \alpha $ and ${\log _e}\left| {\sin \left( {x - \alpha } \right)} \right|$
C.
$x + \alpha $ and ${\log _e}\left| {\sin \left( {x + \alpha } \right)} \right|$
D.
$x - \alpha $ and ${\log _e}\left| {\sin \left( {x - \alpha } \right)} \right|$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th April Morning Slot
The integral $\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx$ is equal to :
(Here C is a constant of integration)
A.
${\log _e}{{\left| {{x^3} + 1} \right|} \over {{x^2}}} + C$
B.
${1 \over 2}{\log _e}{{\left| {{x^3} + 1} \right|} \over {{x^2}}} + C$
C.
${\log _e}\left| {{{{x^3} + 1} \over x}} \right| + C$
D.
${1 \over 2}{\log _e}{{{{\left( {{x^3} + 1} \right)}^2}} \over {\left| {{x^3}} \right|}} + C$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Evening Slot
If $\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c$, where c is a constant of integration, then $g$(–1) is equal to :
A.
1
B.
- 1
C.
$ - {5 \over 2}$
D.
$ - {1 \over 2}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th April Morning Slot
If $\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C$

where C is a constant of integration then :
A.
A =${1 \over {54}}$ and f(x) = 9(x–1)2
B.
A =${1 \over {54}}$ and f(x) = 3(x–1)
C.
A =${1 \over {81}}$ and f(x) = 3(x–1)
D.
A =${1 \over {27}}$ and f(x) = 9(x–1)2
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Evening Slot
$\int {{e^{\sec x}}}$ $(\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx$
= esecxf(x) + C then a possible choice of f(x) is :-
A.
x sec x + tan x + 1/2
B.
sec x + xtan x - 1/2
C.
sec x - tan x - 1/2
D.
sec x + tan x + 1/2
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th April Morning Slot
The integral $\int {{\rm{se}}{{\rm{c}}^{{\rm{2/ 3}}}}\,{\rm{x }}\,{\rm{cose}}{{\rm{c}}^{{\rm{4 / 3}}}}{\rm{x \,dx}}} $ is equal to (Hence C is a constant of integration)
A.
-3/4 tan - 4 / 3 x + C
B.
3tan–1/3x + C
C.
–3cot–1/3x+ C
D.
- 3tan–1/3x + C
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Evening Slot
If $\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C} $
where C is a constant of integration, then the function ƒ(x) is equal to
A.
${3 \over {{x^2}}}$
B.
$ - {1 \over {6{x^3}}}$
C.
$ - {1 \over {2{x^3}}}$
D.
$ - {1 \over {2{x^2}}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 8th April Morning Slot
$\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx} $ is equal to
(where c is a constant of integration)
A.
2x + sinx + 2sin2x + c
B.
x + 2sinx + sin2x + c
C.
x + 2sinx + 2sin2x + c
D.
2x + sinx + sin2x + c
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Evening Slot
The integral $\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$ is equal to : (where C is a constant of integration)
A.
${{{x^{12}}} \over {6{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}}$ + $C$
B.
${{{x^4}} \over {6{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C$
C.
${{{x^{12}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C$
D.
${{{x^4}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 12th January Morning Slot
The integral $\int \, $cos(loge x) dx is equal to : (where C is a constant of integration)
A.
${x \over 2}$[sin(loge x) $-$ cos(loge x)] + C
B.
x[cos(loge x) + sin(loge x)] + C
C.
${x \over 2}$[cos(loge x) + sin(loge x)] + C
D.
x[cos(loge x) $-$ sin(loge x)] + C
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Evening Slot
If   $\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx$ = f(x) $\sqrt {2x - 1} $ + C, where C is a constant of integration, then f(x) is equal to :
A.
${2 \over 3}$ (x $-$ 4)
B.
${1 \over 3}$ (x + 4)
C.
${1 \over 3}$ (x + 1)
D.
${2 \over 3}$ (x + 2)
2019 JEE Mains MCQ
JEE Main 2019 (Online) 11th January Morning Slot
If  $\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}} $ dx = A(x)${\left( {\sqrt {1 - {x^2}} } \right)^m}$ + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))m equals :
A.
${1 \over {27{x^6}}}$
B.
${{ - 1} \over {27{x^9}}}$
C.
${1 \over {9{x^4}}}$
D.
${1 \over {3{x^3}}}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Evening Slot
If  $\int \, $x5.e$-$4x3 dx = ${1 \over {48}}$e$-$4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -
A.
$-$2x3 $-$ 1
B.
$-$ 2x3 + 1
C.
4x3 + 1
D.
$-$4x3 $-$ 1
2019 JEE Mains MCQ
JEE Main 2019 (Online) 10th January Morning Slot
Let n $ \ge $ 2 be a natural number and $0 < \theta < {\pi \over 2}.$ Then $\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}^{1/n}}\cos \theta } \over {{{\sin }^{n + 1}}\theta }}} \,d\theta $ is equal to - (where C is a constant of integration)
A.
${n \over {{n^2} - 1}}{\left( {1 + {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$
B.
${n \over {{n^2} - 1}}{\left( {1 - {1 \over {{{\sin }^{n + 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$
C.
${n \over {{n^2} - 1}}{\left( {1 - {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$
D.
${n \over {{n^2} + 1}}{\left( {1 - {1 \over {{{\sin }^{n - 1}}\theta }}} \right)^{{{n + 1} \over n}}} + C$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Evening Slot
If   $f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),$

$f\left( 0 \right) = 0,$    then the value of $f(1)$ is :
A.
$ - $ ${1 \over 2}$
B.
$ - $ ${1 \over 4}$
C.
${1 \over 2}$
D.
${1 \over 4}$
2019 JEE Mains MCQ
JEE Main 2019 (Online) 9th January Morning Slot
For x2 $ \ne $ n$\pi $ + 1, n $ \in $ N (the set of natural numbers), the integral

$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx} $ is equal to :

(where c is a constant of integration)
A.
${\log _e}\left| {{1 \over 2}{{\sec }^2}\left( {{x^2} - 1} \right)} \right| + c$
B.
${1 \over 2}{\log _e}\left| {\sec \left( {{x^2} - 1} \right)} \right| + c$
C.
${1 \over 2}{\log _e}\left| {{{\sec }^2}\left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$
D.
${\log _e}\left| {\sec \left( {{{{x^2} - 1} \over 2}} \right)} \right| + c$
2018 JEE Mains MCQ
JEE Main 2018 (Online) 16th April Morning Slot
If $\int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}} $ $\left( {{{K\,\tan x + 1} \over {\sqrt A }}} \right) + C,(C\,\,$ is a constant of integration) then the ordered pair (K, A) is equal to :
A.
(2, 1)
B.
($-$2, 3)
C.
(2, 3)
D.
($-$2, 1)
2018 JEE Mains MCQ
JEE Main 2018 (Offline)
The integral

$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx$

is equal to
A.
${{ - 1} \over {1 + {{\cot }^3}x}} + C$
B.
${1 \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$
C.
${{ - 1} \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$
D.
${1 \over {1 + {{\cot }^3}x}} + C$
2018 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Evening Slot
If    $\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \right) + C$
(where C is a constant of integration), then the ordered pair (A, B) is equal to :
A.
(2,  1)
B.
($-$ 2,   $-$1)
C.
($-$ 2,  1)
D.
(2,   $-$1)
2018 JEE Mains MCQ
JEE Main 2018 (Online) 15th April Morning Slot
If $f\left( {{{x - 4} \over {x + 2}}} \right) = 2x + 1,$ (x $ \in $ R $-${1, $-$ 2}), then $\int f \left( x \right)dx$ is equal to :
(where C is a constant of integration)
A.
12 loge | 1 $-$ x | + 3x + C
B.
$-$ 12 loge | 1 $-$ x | $-$ 3x + C
C.
12 loge | 1 $-$ x | $-$ 3x + C
D.
$-$ 12 loge | 1 $-$ x | + 3x + C
2017 JEE Mains MCQ
JEE Main 2017 (Online) 9th April Morning Slot
If $\,\,\,$ f$\left( {{{3x - 4} \over {3x + 4}}} \right)$ = x + 2, x $ \ne $ $-$ ${4 \over 3}$, and

$\int {} $f(x) dx = A log$\left| {} \right.$1 $-$ x $\left| {} \right.$ + Bx + C,

then the ordered pair (A, B) is equal to :

(where C is a constant of integration)
A.
$\left( {{8 \over 3},{2 \over 3}} \right)$
B.
$\left( { - {8 \over 3},{2 \over 3}} \right)$
C.
$\left( { - {8 \over 3}, - {2 \over 3}} \right)$
D.
$\left( { {8 \over 3}, - {2 \over 3}} \right)$
2017 JEE Mains MCQ
JEE Main 2017 (Online) 8th April Morning Slot
The integral

$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} $

$\left( {0 < x < {\pi \over 2}} \right)$ is equal to :

(where C is a constant of integration)
A.
4 log(sin ${x \over 2}$ ) + C
B.
2 log(sin ${x \over 2}$ ) + C
C.
2 log(cos ${x \over 2}$ ) + C
D.
4 log(cos ${x \over 2}$) + C
2017 JEE Mains MCQ
JEE Main 2017 (Offline)
Let ${I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).$

If ${I_4} + {I_6}$ = $a{\tan ^5}x + b{x^5} + C$, where C is a constant of integration,

then the ordered pair $\left( {a,b} \right)$ is equal to
A.
$\left( {{1 \over 5},0} \right)$
B.
$\left( {{1 \over 5}, - 1} \right)$
C.
$\left( { - {1 \over 5},0} \right)$
D.
$\left( { - {1 \over 5},1} \right)$
2016 JEE Mains MCQ
JEE Main 2016 (Online) 10th April Morning Slot
The integral $\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}} $ is equal to :

(where C is a constant of integration.)
A.
$ - 2\sqrt {{{1 + \sqrt x } \over {1 - \sqrt x }}} + C$
B.
$ - 2\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$
C.
$ - \sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}} + C$
D.
$2\sqrt {{{1 + \sqrt x } \over {1 - \sqrt x }}} + C$
2016 JEE Mains MCQ
JEE Main 2016 (Online) 9th April Morning Slot
If   $\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k,$

where k is a constant of integration, then A + B +C equals :
A.
${{21} \over 5}$
B.
${{16} \over 5}$
C.
${{7} \over 10}$
D.
${{27} \over 10}$
2016 JEE Mains MCQ
JEE Main 2016 (Offline)
The integral $\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$ is equal to :
A.
${{{x^5}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$
B.
${{ - {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$
C.
${{{-x^5}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$
D.
${{ {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$
2015 JEE Mains MCQ
JEE Main 2015 (Offline)
The integral $\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $ equals :
A.
$ - {\left( {{x^4} + 1} \right)^{{1 \over 4}}} + c$
B.
$ - {\left( {{{{x^4} + 1} \over {{x^4}}}} \right)^{{1 \over 4}}} + c$
C.
$ {\left( {{{{x^4} + 1} \over {{x^4}}}} \right)^{{1 \over 4}}} + c$
D.
$ {\left( {{x^4} + 1} \right)^{{1 \over 4}}} + c$
2014 JEE Mains MCQ
JEE Main 2014 (Offline)
The integral $\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx} $ is equal to
A.
JEE Main 2014 (Offline) Mathematics - Indefinite Integrals Question 82 English Option 1
B.
JEE Main 2014 (Offline) Mathematics - Indefinite Integrals Question 82 English Option 2
C.
JEE Main 2014 (Offline) Mathematics - Indefinite Integrals Question 82 English Option 3
D.
JEE Main 2014 (Offline) Mathematics - Indefinite Integrals Question 82 English Option 4
2013 JEE Mains MCQ
JEE Main 2013 (Offline)
If $\int {f\left( x \right)dx = \psi \left( x \right),} $ then $\int {{x^5}f\left( {{x^3}} \right)dx} $ is equal to
A.
${1 \over 3}\left[ {{x^3}\psi \left( {{x^3}} \right) - \int {{x^2}\psi \left( {{x^3}} \right)dx} } \right] + C$
B.
${1 \over 3}{x^3}\psi \left( {{x^3}} \right) - 3\int {{x^3}\psi \left( {{x^3}} \right)dx} + C$
C.
${1 \over 3}{x^3}\psi \left( {{x^3}} \right) - \int {{x^2}\psi \left( {{x^3}} \right)dx} + C$
D.
${1 \over 3}\left[ {{x^3}\psi \left( {{x^3}} \right) - \int {{x^3}\psi \left( {{x^3}} \right)dx} } \right] + C$
2012 JEE Mains MCQ
AIEEE 2012
If the $\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\,\ln \,\left| {\sin x - 2\cos x} \right| + k,} $ then $a$ is
equal to :
A.
$-1$
B.
$-2$
C.
$1$
D.
$2$
2012 JEE Advanced MCQ
IIT-JEE 2012 Paper 1 Offline
The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant $K$)
A.
$-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
B.
$\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
C.
$-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
D.
$\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
2008 JEE Mains MCQ
AIEEE 2008
The value of $\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}} $ is
A.
$\,x + \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$
B.
$\,x - \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$
C.
$\,x + \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$
D.
$\,x - \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$
2008 JEE Advanced MCQ
IIT-JEE 2008 Paper 2 Offline
Let $I = \int {{{{e^x}} \over {{e^{4x}} + {e^{2x}} + 1}}dx,\,\,J = \int {{{{e^{ - x}}} \over {{e^{ - 4x}} + {e^{ - 2x}} + 1}}dx.} } $ Then

for an arbitrary constant $C$, the value of $J -I$ equals :
A.
${1 \over 2}\log \left( {{{{e^{4x}} - {e^{2x}} + 1} \over {{e^{4x}} + {e^{2x}} + 1}}} \right) + C$
B.
${1 \over 2}\log \left( {{{{e^{2x}} + {e^x} + 1} \over {{e^{2x}} - {e^x} + 1}}} \right) + C$
C.
${1 \over 2}\log \left( {{{{e^{2x}} - {e^x} + 1} \over {{e^{2x}} + {e^x} + 1}}} \right) + C$
D.
${1 \over 2}\log \left( {{{{e^{4x}} + {e^{2x}} + 1} \over {{e^{4x}} - {e^{2x}} + 1}}} \right) + C$
2007 JEE Mains MCQ
AIEEE 2007
$\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}} $ equals
A.
$\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C$
B.
$\log \,\tan \,\left( {{x \over 2} - {\pi \over {12}}} \right) + C$
C.
$\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C$
D.
$\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} - {\pi \over {12}}} \right) + C$
2007 JEE Advanced MCQ
IIT-JEE 2007
Let $F(x)$ be an indefinite integral of $si{n^2}x.$

STATEMENT-1: The function $F(x)$ satisfies $F\left( {x + \pi } \right) = F\left( x \right)$
for all real $x$. because

STATEMENT-2: ${\sin ^2}\left( {x + \pi } \right) = {\sin ^2}x$ for all real $x$.

A.
Statement-1 is True, Statement-2 is True; Statement-2 is 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.
2007 JEE Advanced MCQ
IIT-JEE 2007 Paper 2 Offline

Let $f(x)=\frac{x}{\left(1+x^{n}\right)^{1 / n}}$ for $n \geq 2$ and $g(x)=\underbrace{(f o f o \ldots . o f)}_{f \text { occurs } n \text { times }}(x)$. Then $\int x^{n-2} g(x) d x$ equals :

A.
$\frac{1}{n(n-1)}\left(1+n x^{n}\right)^{1-\frac{1}{n}}+k$
B.
$\frac{1}{n(n+1)}\left(1+n x^{n}\right)^{1-\frac{1}{n}}+k$
C.
$\frac{1}{n(n-1)}\left(1+n x^{n}\right)^{1-\frac{1}{n}}+k$
D.
$\frac{1}{(n+1)}\left(1+n x^{n}\right)^{1+\frac{1}{n}}+k$
2006 JEE Advanced MCQ
IIT-JEE 2006

$\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x$ is equal to

A.

$\frac{\sqrt{2 x^4-2 x^2+1}}{x^2}+\mathrm{C}$

B.

$\frac{\sqrt{2 x^4-2 x^2+1}}{x^3}+\mathrm{C}$

C.

$\frac{\sqrt{2 x^4-2 x^2+1}}{x}+\mathrm{C}$

D.

$\frac{\sqrt{2 x^4-2 x^2+1}}{2 x^2}+C$

2005 JEE Mains MCQ
AIEEE 2005
$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx} $ is equal to
A.
${{\log x} \over {{{\left( {\log x} \right)}^2} + 1}} + C$
B.
${x \over {{x^2} + 1}} + C$
C.
${{x{e^x}} \over {1 + {x^2}}} + C$
D.
${x \over {{{\left( {\log x} \right)}^2} + 1}} + C$
2005 JEE Advanced MCQ
IIT-JEE 2005 Screening
If $\int\limits_{\sin x}^1 {{t^2}f\left( t \right)dt = 1 - \sin x,} $ then f$\left( {{1 \over {\sqrt 3 }}} \right)$ is
A.
${1 \over 3}$
B.
${{1 \over {\sqrt 3 }}}$
C.
$3$
D.
${\sqrt 3 }$
2004 JEE Mains MCQ
AIEEE 2004
$\int {{{dx} \over {\cos x - \sin x}}} $ is equal to
A.
${1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} + {{3\pi } \over 8}} \right)} \right| + C$
B.
${1 \over {\sqrt 2 }}\log \left| {\cot \left( {{x \over 2}} \right)} \right| + C$
C.
${1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} - {{3\pi } \over 8}} \right)} \right| + C$
D.
$\,{1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} - {\pi \over 8}} \right)} \right| + C$
2004 JEE Mains MCQ
AIEEE 2004
If $\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} $ then value of
$(A, B)$ is
A.
$\left( { - \cos \alpha ,\sin \alpha } \right)$
B.
$\left( { \cos \alpha ,\sin \alpha } \right)$
C.
$\left( { - \sin \alpha ,\cos \alpha } \right)$
D.
$\left( { \sin \alpha ,\cos \alpha } \right)$
2002 JEE Advanced Numerical
IIT-JEE 2002
For any natural number $m$, evaluate
$\int {\left( {{x^{3m}} + {x^{2m}} + {x^m}} \right){{\left( {2{x^{2m}} + 3{x^m} + 6} \right)}^{l/m}}dx,x > 0.} $
2001 JEE Advanced Numerical
IIT-JEE 2001
Evaluate $\int {{{\sin }^{ - 1}}\left( {{{2x + 2} \over {\sqrt {4{x^2} + 8x + 13} }}} \right)} \,dx.$
1999 JEE Advanced Numerical
IIT-JEE 1999
Integrate $\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.} $
1996 JEE Advanced Numerical
IIT-JEE 1996
Evaluate $\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx} $.
1995 JEE Advanced MCQ
IIT-JEE 1995 Screening
The value of the integral $\int {{{{{\cos }^3}x + {{\cos }^5}x} \over {{{\sin }^2}x + {{\sin }^4}x}}} \,dx\,$ is
A.
$\sin x - 6{\tan ^{ - 1}}\left( {\sin x} \right) + c$
B.
$\sin x - 2{\left( {\sin x} \right)^{ - 1}} + c$
C.
$\sin x - 2{\left( {\sin x} \right)^{ - 1}} - 6{\tan ^{ - 1}}\left( {\sin x} \right) + c$
D.
$\,\sin x - 2{\left( {\sin x} \right)^{ - 1}} + 5{\tan ^{ - 1}}\left( {\sin x} \right) + c$