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## MCQ (Single Correct Answer)

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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) 29th July Evening Shift
$$\text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2... JEE Main 2022 (Online) 26th July Evening Shift If$$\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $$,$$g(1) = 0$$, then$$g\left( {{1 \over 2}} \rig...
JEE Main 2022 (Online) 26th June Evening Shift
If $$\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C}$$, where C is a constant, then $${{{d^3}f} \ov... JEE Main 2022 (Online) 27th June Morning Shift The integral$$\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx$$is equal to : (where C is a constant ... JEE Main 2021 (Online) 31st August Morning Shift The integral$$\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx$$is equal to (wh... JEE Main 2021 (Online) 18th March Morning Shift The integral$$\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} -...
JEE Main 2021 (Online) 25th February Evening Shift
The value of the integral $$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\the... JEE Main 2021 (Online) 25th February Morning Slot If$$\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \r...
JEE Main 2021 (Online) 24th February Morning Slot
If $$\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta$$ = A$${\log _e}\left| {B\left( \t... JEE Main 2020 (Online) 5th September Evening Slot If$$\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx} $$=$$g\left...
JEE Main 2020 (Online) 5th 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 in...
JEE Main 2020 (Online) 4th September Morning Slot
If $$\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx$$ = A(x)$${\tan ^{ - 1}}\left( {\sqrt x } \ri... JEE Main 2020 (Online) 3rd September Evening Slot If$$\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta...
JEE Main 2020 (Online) 9th January Evening Slot
The integral $$\int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}}$$ is equal to : (where C is a c...
JEE Main 2020 (Online) 9th January Morning Slot
If ƒ'(x) = tan–1(secx + tanx), $$- {\pi \over 2} &lt; x &lt; {\pi \over 2}$$, and ƒ(0) = 0, then ƒ(1) is equal to :...
JEE Main 2020 (Online) 9th January Morning Slot
If $$\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 ... JEE Main 2020 (Online) 8th January Morning Slot Let$$a \in \left( {0,{\pi \over 2}} \right)$$be fixed. If the integral$$\int {{{\tan x + \tan \alpha } \over {\tan x...
JEE Main 2019 (Online) 12th April Evening 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) 12th April Morning 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 ...
JEE Main 2019 (Online) 10th April Evening Slot
If $$\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}}... JEE Main 2019 (Online) 10th April Morning Slot$$\int {{e^{\sec x}}}(\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx$$= esecxf(x) + C then a possible choice of f(x... JEE Main 2019 (Online) 9th April Evening Slot The integral$$\int {{\rm{se}}{{\rm{c}}^{{\rm{2/ 3}}}}\,{\rm{x }}\,{\rm{cose}}{{\rm{c}}^{{\rm{4 / 3}}}}{\rm{x \,dx}}} $$... JEE Main 2019 (Online) 9th April Morning Slot If$$\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C} $$where C is a constant... JEE Main 2019 (Online) 8th April Evening Slot$$\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx} $$is equal to (where c is a constant of integration) JEE Main 2019 (Online) 8th April Morning Slot The integral$$\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$$is equal to : ... JEE Main 2019 (Online) 12th January Evening Slot The integral$$\int \, $$cos(loge x) dx is equal to : (where C is a constant of integration) JEE Main 2019 (Online) 12th January Morning Slot If &nbsp;&nbsp;$$\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx$$= f(x)$$\sqrt {2x - 1} $$+ C, where C is a constant o... JEE Main 2019 (Online) 11th January Evening Slot If &nbsp;$$\int \, $$x5.e$$-$$4x3 dx =$${1 \over {48}}$$e$$-$$4x3 f(x) + C, where C is a constant of inegration, then ... JEE Main 2019 (Online) 10th January Evening Slot Let n$$ \ge $$2 be a natural number and$$0 &lt; \theta &lt; {\pi \over 2}.$$Then$$\int {{{{{\left( {{{\sin }^n}\t...
JEE Main 2019 (Online) 10th January Morning Slot
If&nbsp;&nbsp;&nbsp;$$f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \... JEE Main 2019 (Online) 9th January Evening Slot For x2$$ \ne $$n$$\pi $$+ 1, n$$ \in $$N (the set of natural numbers), the integral$$\int {x\sqrt {{{2\sin ({x^2}...
JEE Main 2019 (Online) 9th January Morning Slot
If $$\int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}}$$ $$\left( {{{K\,... JEE Main 2018 (Online) 16th April Morning Slot If &nbsp;&nbsp;$$\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}...
JEE Main 2018 (Online) 15th April Evening Slot
If $$f\left( {{{x - 4} \over {x + 2}}} \right) = 2x + 1,$$ (x $$\in$$ R $$-$${1, $$-$$ 2}), then $$\int f \left( x \ri... JEE Main 2018 (Online) 15th April Morning Slot The integral$$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x...
JEE Main 2018 (Offline)
If $$\,\,\,$$ f$$\left( {{{3x - 4} \over {3x + 4}}} \right)$$ = x + 2, x $$\ne$$ $$-$$ $${4 \over 3}$$, and $$\int ... JEE Main 2017 (Online) 9th April Morning Slot The integral$$\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx} \left( {0 &lt; x &lt; {\pi \over 2}} \right...
JEE Main 2017 (Online) 8th April Morning Slot
Let $${I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n &gt; 1} \right).$$ If $${I_4} + {I_6}$$ = $$a{\tan ^5}x + b{x^5} + C... JEE Main 2017 (Offline) The integral$$\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}} $$is equal to : (where C is a con... JEE Main 2016 (Online) 10th April Morning Slot If &nbsp;&nbsp;$$\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} ...
JEE Main 2016 (Online) 9th April Morning Slot
The integral $$\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$$ is equal to :
JEE Main 2016 (Offline)
The integral $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}$$ equals :
JEE Main 2015 (Offline)
The integral $$\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx}$$ is equal to
JEE Main 2014 (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
JEE Main 2013 (Offline)
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 ...
AIEEE 2012
The value of $$\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}}$$ is
AIEEE 2008
$$\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}}$$ equals
AIEEE 2007
$$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx}$$ is ...
AIEEE 2005
$$\int {{{dx} \over {\cos x - \sin x}}}$$ is equal to
AIEEE 2004
If $$\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} ... AIEEE 2004 ## Numerical More If$$\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = } \alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 -...
JEE Main 2021 (Online) 31st August Evening Shift
If $$\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x... JEE Main 2021 (Online) 27th August Evening Shift If$$\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left...
JEE Main 2021 (Online) 27th August 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 $$f(1) = {1 \over ... JEE Main 2021 (Online) 18th March Morning Shift For real numbers$$\alpha$$,$$\beta$$,$$\gamma$$and$$\delta $$, if$$\int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{...
JEE Main 2021 (Online) 16th March Evening Shift

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