JEE Main
Mathematics
Indefinite Integrals
Previous Years Questions

## Numerical

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-...
If $$\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = }$$ $$\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma... If$$\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} $$, where C is a con... If$$\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 ...
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
For real numbers $$\alpha$$, $$\beta$$, $$\gamma$$ and $$\delta$$, if $$\int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)}... ## MCQ (Single Correct Answer) Let$$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)$$is equal to For$$I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x$$, if$$I\left(\frac{\pi}{4}\right)=2^{1011}$$, then$$ \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...
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...
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 :
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)
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...
The integral $$\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx$$,...
The value of the integral $$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta ... If$$\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c$$, where c is a cons... If$$\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta $$= A$${\log _e}\left| {B\left( \theta \right)} \right| + C$$, ... If$$\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} +...
The integral $$\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx}$$ is equal to (where C is a constant of integration):
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 ...
If $$\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( ... If ƒ'(x) = tan–1(secx + tanx),$$ - {\pi \over 2} < x < {\pi \over 2}$$, and ƒ(0) = 0, then ƒ(1) is equal to : The integral$$\int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}} $$is equal to : (where C is a constant of integration) If$$\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...
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)...
The integral $$\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx$$ is equal to : (Here C is a constant of integration)
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 :
If $$\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)...$$\int {{e^{\sec x}}}(\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx$$= esecxf(x) + C then a possible choice of f(x) is :- The integral$$\int {{\rm{se}}{{\rm{c}}^{{\rm{2/ 3}}}}\,{\rm{x }}\,{\rm{cose}}{{\rm{c}}^{{\rm{4 / 3}}}}{\rm{x \,dx}}} $$is equal to (Hence C is a con... If$$\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C} $$where C is a constant of integration, then the func...$$\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx} $$is equal to (where c is a constant of integration) The integral$$\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$$is equal to : (where C is a constant of inte... The integral$$\int \, $$cos(loge x) dx is equal to : (where C is a constant of integration) If$$\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx$$= f(x)$$\sqrt {2x - 1} $$+ C, where C is a constant of integration, then f(x) is eq... If$$\int \, $$x5.e$$-$$4x3 dx =$${1 \over {48}}$$e$$-$$4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -... Let n$$ \ge $$2 be a natural number and$$0 < \theta < {\pi \over 2}.$$Then$$\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}...
If   $$f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right)... For x2$$ \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...
If $$\int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}}$$ $$\left( {{{K\,\tan x + 1} \over {\sqrt A }}}... The integral$$\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} \...
If    $$\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \ri... If$$f\left( {{{x - 4} \over {x + 2}}} \right) = 2x + 1,$$(x$$ \in $$R$$-$${1,$$-$$2}), then$$\int f \left( x \right)dx$$is equal to : (where ... If$$\,\,\,$$f$$\left( {{{3x - 4} \over {3x + 4}}} \right)$$= x + 2, x$$ \ne -{4 \over 3}$$, and$$\int {} $$f(x) dx = A log$$\left| {...
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...
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...
The integral $$\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}}$$ is equal to : (where C is a constant of integration.) ...
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...
The integral $$\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$$ is equal to :
The integral $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}$$ equals :
The integral $$\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx}$$ is equal to
If $$\int {f\left( x \right)dx = \psi \left( x \right),}$$ then $$\int {{x^5}f\left( {{x^3}} \right)dx}$$ is equal to
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 :
The value of $$\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}}$$ is
$$\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}}$$ equals
$$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx}$$ is equal to
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...
$$\int {{{dx} \over {\cos x - \sin x}}}$$ is equal to
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