$\smallint \,\,{\left( {\frac{{x + 2}}{{x + 4}}} \right)^2}{e^x}\,\;dx$ is equal to

${e^x}\left( {\frac{{x + 2}}{{x + 4}}} \right) + c$

${e^x}\left( {\frac{{x - 2}}{{x + 4}}} \right) + c$

$\left( {\frac{{2x{e^x}}}{{x + 4}}} \right) + c$

$\smallint \frac{{\sin \,\,2x}}{{1\,\, + \,\,{{\cos }^2}\,\,x}}\,\,dx$ is equal to

$ - \frac{1}{2}\,\,\log \,\,\left( {1\,\, + \,\,{{\cos }^2}x} \right) + c$

$2\,\,\log \,\,\left( {1\,\, + \,\,{{\cos }^2}x} \right) + c$

$\frac{1}{2}\,\,\log \,\,\left( {1\,\, + \,\,\cos \,\,\,2x} \right) + c$

If $\int \frac{1}{1-\sin ^4 x} d x=\frac{1}{2} \tan x+A \tan ^{-1}\{f(x)\}+C$, then

$A=\sqrt{2}$ and $f(x)=\sqrt{2} \tan x$

$A=-\sqrt{2}$ and $f(x)=\sqrt{2} \tan x$

The integral $\int {{{\sec }^{2/3}}x\,{\rm{cose}}{{\rm{c}}^{4/3}}x\,dx} $ is equal to (here $C$ is a constant of integration)

$ - \frac{3}{4}{\tan ^{ - 4/3}}x + C$

$\smallint \,\,\frac{{{x^4} - \,\,1}}{{{x^2}\,\sqrt {{x^4} + \,\,{x^2} + 1} }}\,\,dx$ is equal to

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

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

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

If an anti-derivate of $f\,\,\left( x \right)$ is e^x and that of ${\rm{g}}\,\,\left( x \right)$ is cos⁡x, then $\smallint \,\,f\left( x \right)\,\,\cos \,\,x\,\,dx\,\, + \,\,\smallint \,\,{\rm{g}}\,\,\left( x \right)\,\,{e^x}\,dx$ is equal to

$f\,\,\left( x \right){\rm{g}}\,\,\left( x \right)\,\, + \,\,c$

$f\,\,\left( x \right)\,\, + \,\,{\rm{g}}\,\,\left( x \right)\,\, + \,\,c$

$f\,\,\left( x \right)\,\, - \,\,{\rm{g}}\,\,\left( x \right)\,\, + \,\,c$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Easy

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 361-390 of 746 questions

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361 Easy

361 of 746

Analyzing +4 | -1

\int \,\,{\left( {\frac{{x + 2}}{{x + 4}}} \right)^2}{e^x}\,\;dx

is equal to

View answer and explanation

Correct option: A

{e^x}\left( {\frac{x}{{x + 4}}} \right) + c

Option A (

{e^x}\left( {\frac{x}{{x + 4}}} \right) + c

) is correct because the integral can be solved by manipulating the integrand and using integration by parts.

362 Easy

362 of 746

Application +4 | -1

The value of

\int {\frac{{{{\cos }^3}xdx}}{{{{\sin }^2}x + \sin x}}}

is equal to

View answer and explanation

Correct option: B

\log \left| {\sin x} \right| - \sin x + c

Option B (

\log \left| {\sin x} \right| - \sin x + c

) is correct because after substituting

t = \sin x

and simplifying, the integral evaluates to this expression.

363 Easy

363 of 746

Application +4 | -1

\int x e^x \cos x d x=f(x)+c

, then

f(x)

is equal to

View answer and explanation

Correct option: A

\frac{e^x}{2}\{(1-x) \sin x-x \cos x\}

Option A (

\frac{e^x}{2}\{(1-x) \sin x-x \cos x\}

) is correct because it is the result of integrating

xe^x \cos x

using complex numbers and extracting the real part.

364 Easy

364 of 746

Application +4 | -1

\int \frac{{\sin \,\,2x}}{{1\,\, + \,\,{{\cos }^2}\,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: D

c\,\, - \,\,\log \,\,\left( {1\,\, + \,\,{{\cos }^2}\,\,x} \right)

Option D,

c\,\, - \,\,\log \,\,\left( {1\,\, + \,\,{{\cos }^2}\,\,x} \right)

, is correct because it is the result of integrating the given trigonometric function using a simple substitution.

365 Easy

365 of 746

Application +4 | -1

\int f(x) d x=\phi(x)

i.e.

\phi

is a function such that

\phi^{\prime}(x)

=f(x)

, then

\int x^9 f\left(x^5\right) d x

is equal to

View answer and explanation

Correct option: C

\frac{1}{5} x^5 \phi\left(x^5\right)-\int x^4 \phi\left(x^5\right) d x+C

Option C (

\frac{1}{5} x^5 \phi(x^5)-\int x^4 \phi(x^5) d x+C

) is correct because it is derived by using substitution and integration by parts.

366 Easy

366 of 746

Application +4 | -1

If the integral

\int \frac{5 \tan x}{\tan x-2} d x=x+a \log

|\sin x-2 \cos x|+k

, then

a

is equal to

View answer and explanation

Correct option: D

Option D (2) is correct because it represents the value of 'a' obtained by comparing the given integral with the provided expression after solving the integral.

367 Easy

367 of 746

Application +4 | -1

\int \frac{\left(x^2-1\right) \mathrm{d} x}{\left(x^4+3 x^2+1\right) \tan ^{-1} \frac{x^2+1}{x}}=

\log \left|\tan ^{-1} f(x)\right|+C

then

View answer and explanation

Correct option: D

f(x)=\frac{x^2+1}{x}

Option D (

f(x)=\frac{x^2+1}{x}

) is correct because after dividing the numerator and denominator by

x^2

and using appropriate substitutions, the integral simplifies to

\log |\tan^{-1} (\frac{x^2+1}{x})| + C

368 Easy

368 of 746

Analyzing +4 | -1

\int \frac{1}{1-\sin ^4 x} d x=\frac{1}{2} \tan x+A \tan ^{-1}\{f(x)\}+C

, then

View answer and explanation

Correct option: A

A=\frac{1}{2 \sqrt{2}}

and

f(x)=\sqrt{2} \tan x

Option A is correct because after simplifying the integral using trigonometric identities and partial fractions, the given values for A and f(x) are obtained.

369 Easy

369 of 746

Application +4 | -1

\int\limits_{{-1}}^1 x^2 \sin 2 x d x=

View answer and explanation

Correct option: B

-\frac{1}{2} x^2 \cos 2 x+\frac{1}{2} x \sin 2 x+\frac{1}{4} \cos 2 x+c

Option B,

-\frac{1}{2} x^2 \cos 2 x+\frac{1}{2} x \sin 2 x+\frac{1}{4} \cos 2 x+c

, is the correct antiderivative, but the question asks for the definite integral from -1 to 1 of an odd function, which is 0.

370 Easy

370 of 746

Understanding +4 | -1

\int \,\,\frac{{f'\,\,\left( x \right)}}{{\sqrt {f\,\,\left( x \right)} }}\,\,dx = ...\,\, + \,\,c\,;\,\,f\left( x \right) \ne 0

View answer and explanation

Correct option: B

2\,\,\sqrt {f\,\,\left( x \right)}

Option B (

2\,\,\sqrt {f\,\,\left( x \right)}

) is correct because the integral of

\frac{{f'\,\,\left( x \right)}}{{\sqrt {f\,\,\left( x \right)} }}

2\sqrt{f(x)} + c

371 Easy

371 of 746

Understanding +4 | -1

\int \frac{{f'\left( x \right)}}{{f\left( x \right)\,\,\log \,\,\left[ {f\left( x \right)} \right]}}\,\,dx

is equal to

View answer and explanation

Correct option: C

\log \,\,\left[ {\log \,\,f\left( x \right)} \right]\,\, + \,\,c

Option C (log[log f(x)] + c) is correct because it is the result of applying u-substitution twice, first with u = log(f(x)) and then with v = log(u).

372 Easy

372 of 746

Application +4 | -1

\int \frac{\sin ^4 x}{\cos ^8 x} d x=a \tan ^7 x+b \tan ^5 x+C

, then

View answer and explanation

Correct option: A

7 a=5 b

Option B is correct because after simplifying the integral using trigonometric identities and integration, we find 5a = 7b.

373 Easy

373 of 746

Application +4 | -1

The integral

\int {{{\sec }^{2/3}}x\,\mathrm{cose}{\mathrm{c}^{4/3}}x\,dx}

is equal to (here

C

is a constant of integration)

View answer and explanation

Correct option: B

- 3{\tan ^{ - 1/3}}x + C

Option B (

- 3{\tan ^{ - 1/3}}x + C

) is correct because the integral evaluates to this expression after substitution and simplification.

374 Easy

374 of 746

Analyzing +4 | -1

Let

I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x,

Then, for an arbitrary constant

C

, the value of

J-I

equals

View answer and explanation

Correct option: C

\frac{1}{2} \log \left(\frac{e^{2 x}-e^x+1}{e^{2 x}+e^x+1}\right)+C

Option C (

\frac{1}{2} \log \left(\frac{e^{2 x}-e^x+1}{e^{2 x}+e^x+1}\right)+C

) is correct because after simplifying the integrals and using substitution, the result matches this option.

375 Easy

375 of 746

Analyzing +4 | -1

The value of

I=\int \frac{d x}{\left(a+b x^2\right) \sqrt{b-a x^2}}

, is

View answer and explanation

Correct option: A

\frac{1}{\sqrt{a\left(a^2+b^2\right)}} \tan ^{-1}\left(\frac{x \sqrt{a^2+b^2}}{a \sqrt{b-a x^2}}\right)+C

Option A (

\frac{1}{\sqrt{a(a^2+b^2)}} \tan ^{-1}\left(\frac{x \sqrt{a^2+b^2}}{a \sqrt{b-a x^2}}\right)+C

) is correct because the integral simplifies to this inverse tangent form after trigonometric substitution.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

376 Easy

376 of 746

Application +4 | -1

\int \cos \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2} x^2+k

Option B (

\frac{1}{2} x^2+k

) is correct because by substituting

x = \cos 2\theta

and simplifying the integral, we arrive at the given result.

377 Easy

377 of 746

Application +4 | -1

\int \tan ^3 2 x \sec 2 x d x=

View answer and explanation

Correct option: A

\frac{1}{6} \sec ^3 2 x-\frac{1}{2} \sec 2 x+c

Option A (

\frac{{1}}{{6}} \sec ^3 2 x-\frac{{1}}{{2}} \sec 2 x+c

) is correct because the integral can be solved by substitution and using trigonometric identities.

378 Easy

378 of 746

Application +4 | -1

\int \frac{e^x d x}{\sqrt{1-e^{2 x}}}=

View answer and explanation

Correct option: B

-\cos ^{-1}\left(e^x\right)+c

Option B (

-\cos ^{-1}\left(e^x\right)+c

) is correct because it is the result of a u-substitution and applying the standard integral of

\frac{1}{\sqrt{1-t^2}}

379 Easy

379 of 746

Analyzing +4 | -1

\int \,\,\frac{{{x^4} - \,\,1}}{{{x^2}\,\sqrt {{x^4} + \,\,{x^2} + 1} }}\,\,dx

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because after simplifying and integrating the given expression, the result matches this form.

380 Easy

380 of 746

Application +4 | -1

{I}=\int \frac{d x}{x^3\left(x^8+1\right)^{3 / 4}}=\frac{\lambda\left(1+x^8\right)^{\frac{1}{4}}}{x^2}+c

(where

c

is the constant of integration), then the value of

\lambda

is equal to

View answer and explanation

Correct option: D

-\frac{1}{2}

Option D (-

\frac{{1}}{{2}}

) is correct because it is the value of

\lambda

obtained after solving the integral.

381 Easy

381 of 746

Application +4 | -1

\int x^5 \sqrt[3]{\left(1+x^3\right)^2} d x

is equal to

View answer and explanation

Correct option: A

\frac{1}{8} u^{8 / 3}-\frac{1}{5} u^{5 / 3}+C, u=1+x^3

Option A (

\frac{1}{8} u^{8 / 3}-\frac{1}{5} u^{5 / 3}+C, u=1+x^3

) is correct because after the substitution

u = 1 + x^3

, the integral simplifies and can be easily integrated.

382 Easy

382 of 746

Application +4 | -1

If an anti-derivate of

f\,\,\left( x \right)

is e^x and that of

\mathrm{g}\,\,\left( x \right)

is cos⁡x, then

\int \,\,f\left( x \right)\,\,\cos \,\,x\,\,dx\,\, + \,\,\int \,\,\mathrm{g}\,\,\left( x \right)\,\,{e^x}\,dx

is equal to

View answer and explanation

Correct option: C

{e^x}\,\,\cos \,\,x\,\, + \,\,c

Option C (

{e^x}\,\,\cos \,\,x\,\, + \,\,c

) is correct because integrating by parts and simplifying leads to this result.

383 Easy

383 of 746

Application +4 | -1

The integral

\int {\cos \left( {{{\log }_e}x} \right)dx}

is equal to (where C is a constant of integration)

View answer and explanation

Correct option: A

\frac{x}{2}\left[ {\cos \left( {{{\log }_e}x} \right) + \sin \left( {{{\log }_e}x} \right)} \right] + C

Option A is correct. The integral is solved using integration by parts twice, leading to the expression (x/2)[cos(log_e(x)) + sin(log_e(x))] + C.

384 Easy

384 of 746

Application +4 | -1

\int \frac{\mathrm{d} x}{\sqrt{2-3 x-x^2}}=f o g(x)+C

then

View answer and explanation

Correct option: C

f(x)=\sin ^{-1} x, g(x)=\frac{2 x+3}{\sqrt{17}}

Option C (

f(x)=\sin ^{-1} x, g(x)=\frac{{2 x+3}}{{\sqrt{{17}}}}

) is correct because completing the square and using the substitution

u = \frac{{2x+3}}{{\sqrt{{17}}}}

leads to the arcsine integral.

385 Easy

385 of 746

Analyzing +4 | -1

\int x^2 \frac{\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x=

View answer and explanation

Correct option: B

\frac{-x^2}{(x \tan x+1)}+2 \ln |x \sin x+\cos x|+C

Option B (

\frac{-x^2}{(x \tan x+1)}+2 \ln |x \sin x+\cos x|+C

) is correct because integrating by parts and simplifying leads to this result.

386 Easy

386 of 746

Analyzing +4 | -1

\int \frac{e^{\left(x^2+4 \ln x\right)}-x^3 e^{x^2}}{x-1} d x

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to

\frac{1}{2}(x^2-1) e^{x^2}+C

, which is not among the given options.

387 Easy

387 of 746

Application +4 | -1

\int\left(\frac{x^2-x-1}{x}+1\right)^{K+7}\left(\frac{x^2+1}{x^2}\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{\left(x-\frac{1}{x}\right)^{K+8}}{k+8}+c

Option B (

\frac{(x-\frac{1}{x})^{K+8}}{k+8}+c

) is correct because substituting

t = x - \frac{1}{x}

simplifies the integral to a basic power rule integration.

388 Easy

388 of 746

Application +4 | -1

\int \frac{\cos 2 x}{\cos x+\sin x} d x

is equal to

View answer and explanation

Correct option: C

\sin x+\cos x+c

Option C (

\sin x+\cos x+c

) is correct because the integral simplifies to

\int (\cos x - \sin x) dx

, which results in

\sin x + \cos x + c

389 Easy

389 of 746

Analyzing +4 | -1

\int \frac{\sin ^3 x}{\left(\cos ^4 x+3 \cos ^2 x+1\right) \tan ^{-1}(\sec x+\cos x)} d x

is equal to

View answer and explanation

Correct option: B

\log _e\left|\tan ^{-1}(\sec x+\cos x)\right|+C

Option B is correct because the integral simplifies to a logarithmic form involving the inverse tangent function.

390 Easy

390 of 746

Analyzing +4 | -1

Let

\int \frac{\cos x}{\sin ^3 x\left(1+\sin ^6 x\right)^{2 / 3}} d x=f(x)\left(1+\sin ^6 x\right)^{1 / \lambda}+C.

Then

\lambda f\left(\frac{\pi}{3}\right)=

View answer and explanation

Correct option: B

-2

Option B (-2) is correct because after simplifying and substituting, the value of

\lambda f(\frac{\pi}{3})

is -2.

Revision notes and key takeaways

Review INDEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

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