# Indefinite Integrals · Mathematics · JEE Main

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JEE Main 2024 (Online) 9th April Morning Shift
Let $$\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C$$, where $$C$$ is the consta...
JEE Main 2024 (Online) 8th April Morning Shift
Let $$I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x$$. If $$I(0)=3$$, then $$I\left(\frac{\pi}{12}\right)$$ is equal to
JEE Main 2024 (Online) 6th April Evening Shift
If $$\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$$ constant, then the maximum value ...
JEE Main 2024 (Online) 29th January Evening Shift
If $$\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B... JEE Main 2024 (Online) 29th January Morning Shift For$$x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, if$$y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin...
JEE Main 2024 (Online) 27th January Evening Shift
$$\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { i... JEE Main 2023 (Online) 10th April Evening Shift For$$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if$$\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x ...
JEE Main 2023 (Online) 10th April Morning Shift
If $$I(x) = \int {{e^{{{\sin }^2}x}}(\cos x\sin 2x - \sin x)dx}$$ and $$I(0) = 1$$, then $$I\left( {{\pi \over 3}} \right)$$ is equal to :
JEE Main 2023 (Online) 8th April Evening Shift
The integral $$\int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x$$ is equal to :
JEE Main 2023 (Online) 8th April Morning Shift
Let $$I(x)=\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x, x > 0$$. If $$\lim_\limits{x \rightarrow \infty} I(x)=0$$, then $$I(1)$$ is equal to :...
JEE Main 2023 (Online) 6th April Morning Shift
Let $$I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x$$. If $$I(0)=0$$, then $$I\left(\frac{\pi}{4}\right)$$ is equal to...
JEE Main 2023 (Online) 25th January Morning Shift
Let $$f(x) = \int {{{2x} \over {({x^2} + 1)({x^2} + 3)}}dx}$$. If $$f(3) = {1 \over 2}({\log _e}5 - {\log _e}6)$$, then $$f(4)$$ is equal to
JEE Main 2022 (Online) 29th July Evening Shift
For $$I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$$, if $$I\left(\frac{\pi}{4}\right)=2^{1011}$$, then
JEE Main 2022 (Online) 26th July Evening Shift
$$\text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x \text { is equa... JEE Main 2022 (Online) 27th June Morning Shift If$$\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C} $$, where C is a constant, then$${{{d^3}f} \over {d{x^3}}}$$at x = 1 is equ... JEE Main 2022 (Online) 26th June Evening Shift If$$\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $$,$$g(1) = 0$$, then$$g\left( {{1 \over 2}} \right)$$is equal to : JEE Main 2021 (Online) 31st August Morning Shift The integral$$\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx$$is equal to : (where C is a constant of integration) JEE Main 2021 (Online) 18th March Morning Shift The integral$$\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx$$is equal to (where c is a constant of integra... JEE Main 2021 (Online) 25th February Evening Shift The integral$$\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx$$,... JEE Main 2021 (Online) 25th February Morning Shift The value of the integral$$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta ...
JEE Main 2021 (Online) 24th February Morning Shift
If $$\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c$$, where c is a cons...
JEE Main 2020 (Online) 5th September Evening Slot
If $$\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta$$ = A$${\log _e}\left| {B\left( \theta \right)} \right| + C$$, ...
JEE Main 2020 (Online) 5th September Morning Slot
If $$\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx}$$ = $$g\left( x \right){e^{\left( {{e^x} +... JEE Main 2020 (Online) 4th September Morning Slot Let$$f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)} $$. Then f(3) – f(1) is eqaul to : JEE Main 2020 (Online) 4th September Morning Slot The integral$$\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx} $$is equal to (where C is a constant of integration): JEE Main 2020 (Online) 3rd September Evening Slot If$$\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx$$= A(x)$${\tan ^{ - 1}}\left( {\sqrt x } \right)$$+ B(x) + C, where C is ... JEE Main 2020 (Online) 9th January Evening Slot If$$\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( ...
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 :
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)
JEE Main 2020 (Online) 8th January Morning Slot
If $$\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\la... JEE Main 2019 (Online) 12th April Evening Slot Let$$a \in \left( {0,{\pi \over 2}} \right)$$be fixed. If the integral$$\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx$$= A(x)... JEE Main 2019 (Online) 12th April Morning Slot The integral$$\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx$$is equal to : (Here C is a constant of integration) JEE Main 2019 (Online) 10th April Evening Slot If$$\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c$$, where c is a constant of integration, then$$g$$(–1) is equal to : JEE Main 2019 (Online) 10th April Morning Slot If$$\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)...
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 :-
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 con...
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 func...
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)
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 inte...
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)
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 eq...
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...
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 -...
JEE Main 2019 (Online) 10th January Morning Slot
Let n $$\ge$$ 2 be a natural number and $$0 < \theta < {\pi \over 2}.$$ Then $$\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}... JEE Main 2019 (Online) 9th January Evening Slot If$$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)...
JEE Main 2019 (Online) 9th January Morning Slot
For x2 $$\ne$$ n$$\pi$$ + 1, n $$\in$$ N (the set of natural numbers), the integral $$\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \ov... JEE Main 2018 (Online) 16th April Morning Slot If$$\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 }}}...
JEE Main 2018 (Offline)
The integral $$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \... JEE Main 2018 (Online) 15th April Evening Slot If$$\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \ri...
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 ...
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$$\left| {... JEE Main 2017 (Online) 8th April Morning Slot The integral$$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} \left( {0 < x < {\pi \over 2}} \right)$$is equal to : (where C is... JEE Main 2017 (Offline) Let$${I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).$$If$${I_4} + {I_6}$$=$$a{\tan ^5}x + b{x^5} + C$$, where C is a constant of in... JEE Main 2016 (Online) 10th April Morning Slot The integral$$\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}} $$is equal to : (where C is a constant of integration.) ... JEE Main 2016 (Online) 9th April Morning Slot If$$\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k,$$where k is a... JEE Main 2016 (Offline) The integral$$\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$$is equal to : JEE Main 2015 (Offline) The integral$$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $$equals : JEE Main 2014 (Offline) The integral$$\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx} $$is equal to JEE Main 2013 (Offline) If$$\int {f\left( x \right)dx = \psi \left( x \right),} $$then$$\int {{x^5}f\left( {{x^3}} \right)dx} $$is equal to AIEEE 2012 If the$$\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\,\ln \,\left| {\sin x - 2\cos x} \right| + k,} $$then$$a$$is equal to : AIEEE 2008 The value of$$\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}} $$is AIEEE 2007$$\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}} $$equals AIEEE 2005$$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx} $$is equal to AIEEE 2004 If$$\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} $$then value of$$(A, B)$$i... AIEEE 2004$$\int {{{dx} \over {\cos x - \sin x}}} $$is equal to ## Numerical JEE Main 2024 (Online) 8th April Evening Shift If$$\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}$$, where$$\mathrm{C}$$is... JEE Main 2024 (Online) 4th April Evening Shift If$$\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _x\left|\tan \...
JEE Main 2023 (Online) 15th April Morning Shift
Let $f(x)=\int \frac{d x}{\left(3+4 x^{2}\right) \sqrt{4-3 x^{2}}},|x| and$f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right)$, ... JEE Main 2023 (Online) 12th April Morning Shift Let $$I(x)=\int \sqrt{\frac{x+7}{x}} \mathrm{~d} x$$ and $$I(9)=12+7 \log _{e} 7$$. If $$I(1)=\alpha+7 \log _{e}(1+2 \sqrt{2})$$, then $$\alpha^{4}$$ ... JEE Main 2023 (Online) 30th January Evening Shift If$\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$constant, then$\beta-...
JEE Main 2021 (Online) 31st August Evening Shift
If $$\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = }$$ $$\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma... JEE Main 2021 (Online) 27th August Evening Shift If$$\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C} $$, where C is a con... JEE Main 2021 (Online) 27th August Morning Shift If$$\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left( {{{2x + 1} \over {{x^2} + x ...
JEE Main 2021 (Online) 18th March Morning Shift
If $$f(x) = \int {{{5{x^8} + 7{x^6}} \over {{{({x^2} + 1 + 2{x^7})}^2}}}dx,(x \ge 0),f(0) = 0}$$ and $$f(1) = {1 \over K}$$, then the value of K is
JEE Main 2021 (Online) 16th March Evening Shift
For real numbers $$\alpha$$, $$\beta$$, $$\gamma$$ and $$\delta$$, if \int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)}...
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