$\smallint \,\,\frac{{x\,\,dx}}{{{x^2}\,\, + \,\,4x\,\, + \,\,5}}$ is equal to

$\frac{1}{2}\,\,\log \,\,\left( {{x^2}\,\, + \,\,4x\,\, + \,\,5} \right)\,\, + \,\,2\,\,{\tan ^{ - 1}}\,\,\left( x \right) + c$

$\smallint \,\,\frac{{\left( {x + 1} \right)}}{{x{{\left( {1 + x{e^x}} \right)}^2}}}\,\,dx$ is

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

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

$\frac{1}{\sqrt{2}} \operatorname{Tan}^{-1}(\sqrt{2} \tan x)$

$\frac{1}{2 \sqrt{2}} \operatorname{Tan}^{-1}(\sqrt{2} \sin x)$

$\smallint \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\,\sqrt {{x^2} - \,\,1} }}\,\,dx$ is equal to

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

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

If $f\left( x \right) = \cos \,\,x - {\cos ^2}\,\,x\,\, + \,\,{\cos ^3}\,\,x - \,\,.\,..\infty $, then $\smallint f\left( x \right)dx$ equals

$x\,\, - \,\,\frac{1}{2}\tan \,\,\frac{x}{2} + c$

$\smallint \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\sqrt {{x^2} - \,\,1} }}\;\,\,dx$ is equal to

$\frac{{{x^2}}}{{\sqrt {{x^2} - \,\,1} }}\, + c$

$ - \,\,\frac{{{x^2}}}{{\sqrt {{x^2} - \,\,1} }}\, + \,c$

$\smallint \frac{{{{\left( {{x^4} - \,\,x} \right)}^{1/4}}}}{{{x^5}}}\,\,dx$ is equal to

$\frac{4}{5}\,\,{\left( {1\,\, - \,\,\frac{1}{{{x^3}}}} \right)^{5/4}} + \,\,C$

$\smallint \,\,\frac{{\cos \,\,x\,\, + \,\,x\,\,\sin \,\,x}}{{x\,\,\left( {x\,\, + \,\,\cos \,\,x} \right)}}\,\,dx$ is equal to

$\log \,\,\left( {x\,\,\left( {x\,\, + \,\,\cos \,\,x} \right)} \right) + C$

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

$\smallint \,\,\frac{1}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}\,\,dx$ is equal to

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\cot \,\,\frac{x}{2}\,} \right| + C$

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, - \,\,\frac{\pi }{8}} \right)\,} \right| + C$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

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Practice format and chapter quiz

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 411 questions

361 Medium

361 of 411

Application +4 | -1

\int \,\,\frac{{x\,\,dx}}{{{x^2}\,\, + \,\,4x\,\, + \,\,5}}

is equal to

View answer and explanation

Correct option: D

\frac{1}{2}\,\,\log \,\,\left( {{x^2}\,\, + \,\,4x\,\, + \,\,5} \right)\,\, - \,\,2\,\,{\tan ^{ - 1}}\,\,\left( {x + 2} \right) + c

Option B (

\frac{1}{2}\,\,\log \,\,\left( {{x^2}\, + \,4x\, + \,5} \right)\, - \,{\tan ^{ - 1}}\,\,\left( {x + 2} \right) + c

) is correct. Completing the square and splitting the integral leads to the correct result.

362 Medium

362 of 411

Understanding +4 | -1

Assertion :

\int[\sin (\log x)+\cos (\log x)] d x=x \sin (\log x)+C

Reason :

\frac{d}{d x}[x \sin (\log x)]=\sin (\log x)+\cos (\log x)

View answer and explanation

Correct option: A

Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.

Option A is correct because the assertion is true, and the reason correctly explains why the assertion is true. The derivative of

x \sin(\log x)

is indeed

\sin(\log x) + \cos(\log x)

, which validates the integral.

363 Medium

363 of 411

Application +4 | -1

Let

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

then which of the following is/are correct ?

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 the integral evaluates to

\sin^{-1}(\frac{2x+3}{\sqrt{17}}) + C

364 Medium

364 of 411

Application +4 | -1

\int \,\,{\left( {\sin \,\,x\,\, - \,\,\cos \,\,x} \right)^4}\,\,\left( {\sin \,\,x\,\, + \,\,\cos \,\,x} \right)\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{{{\left( {\sin \,\,x\,\, - \,\,\cos \,\,x} \right)}^5}}}{5}\,\, + \,\,c

Option B (

\frac{{{{\left( {\sin \\ x\\, - \\,\cos \\ x} \right)}^5}}}{5}\\ + \\,c

) is correct because the integral can be solved by direct substitution.

365 Medium

365 of 411

Application +4 | -1

\int \sin \left(2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x=A \sin ^{-1} x+B x \sqrt{1-x^2}+C

, then

\mathrm{A}+\mathrm{B}=

View answer and explanation

Correct option: A

Option A (1) is correct because the integral simplifies to a known form, and the coefficients of the inverse sine and the term with the square root add up to 1.

366 Medium

366 of 411

Application +4 | -1

\int \frac{x-2}{x^2-4 x+3} d x=

View answer and explanation

Correct option: A

\log \sqrt{x^2-4 x+3}+c

Option A is correct because the integral simplifies to

\log \sqrt{x^2-4 x+3}+c

after substitution.

367 Medium

367 of 411

Analyzing +4 | -1

\int \,\,\frac{{\left( {x + 1} \right)}}{{x{{\left( {1 + x{e^x}} \right)}^2}}}\,\,dx

View answer and explanation

Correct option: A

\log \,\,\left( {\frac{{x{e^x}}}{{x{e^x} + 1}}} \right)\,\, + \,\,\frac{1}{{1 + x{e^x}}}\,\, + \,\,c

Option A is correct because the integral simplifies to a logarithmic form plus a rational term after substitution and partial fraction decomposition.

368 Medium

368 of 411

Analyzing +4 | -1

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

then

f(x)

View answer and explanation

Correct option: C

\frac{1}{2 \sqrt{2}} \operatorname{Tan}^{-1}(\sqrt{2} \tan x)

Option C is correct because the integral can be solved by trigonometric manipulation and substitution, leading to the given expression for f(x).

369 Medium

369 of 411

Application +4 | -1

\int \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\,\sqrt {{x^2} - \,\,1} }}\,\,dx

is equal to

View answer and explanation

Correct option: D

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

Option D is correct because after simplifying and using trigonometric substitution, the integral evaluates to

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

370 Medium

370 of 411

Analyzing +4 | -1

\int \frac{{{{\sin }^3}\,\,x}}{{\left( {1\,\, + \,\,{{\cos }^2}\,\,x} \right)\sqrt {1\,\, + \,\,{{\cos }^2}\,\,x\,\, + \,\,{{\cos }^4}\,\,x} }}\,\,dx

is equal to

View answer and explanation

Correct option: A

{\sec ^{ - 1}}\,\,\left( {\sec \,\,x + \cos \,\,x} \right) + C

Option A (

{\sec ^{ - 1}}\,\,\left( {\sec \,x + \cos \,x} \right) + C

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

371 Medium

371 of 411

Application +4 | -1

\int \frac{\sqrt{5+x^{10}}}{x^{16}} d x

is equal to

View answer and explanation

Correct option: A

-\frac{1}{75}\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+c

Option A (

\frac{-1}{75}(1+\frac{5}{x^{10}})^{3 / 2}+c

) is correct because it is the result of integrating the given expression using substitution.

372 Medium

372 of 411

Application +4 | -1

\int \frac{d x}{4 \sin ^2 x+5 \cos ^2 x}=

View answer and explanation

Correct option: C

\frac{1}{2 \sqrt{5}} \tan ^{-1}\left(\frac{2 \tan x}{\sqrt{5}}\right)+c

Option C (

\frac{{1}}{{2 \sqrt{{5}}}} \tan^{{-1}}\left( \frac{{2 \tan x}}{{\sqrt{{5}}}} \right) + c

) is correct because it is the result of integrating the given expression after trigonometric manipulation and substitution.

373 Medium

373 of 411

Analyzing +4 | -1

The value of

\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^3 x+\cos ^3 x\right)^2} d x

, is

View answer and explanation

Correct option: B

-\frac{1}{3\left(1+\tan ^3 x\right)}

Option B (

-\frac{1}{3\left(1+\tan ^3 x\right)}

) is correct because by dividing by

\cos^6 x

and using substitution, the integral simplifies to a form that can be easily integrated.

374 Medium

374 of 411

Application +4 | -1

u=\int e^{a x} \sin b x \,d x

and

v=\int e^{a x} \cos b x \,d x

, then

\tan ^{-1}\left(\frac{u}{v}\right)+\tan ^{-1}\left(\frac{b}{a}\right)

equals

View answer and explanation

Correct option: A

b x

Option A,

b x

, is correct because by expressing

u

and

v

in terms of trigonometric functions and using the arctangent addition formula, the expression simplifies to

bx

375 Medium

375 of 411

Application +4 | -1

f\left( x \right) = \cos \,\,x - {\cos ^2}\,\,x\,\, + \,\,{\cos ^3}\,\,x - \,\,.\,..\infty

, then

\int f\left( x \right)dx

equals

View answer and explanation

Correct option: D

x\,\, - \,\,\tan \,\,\frac{x}{2} + c

Option D (

x - \tan \frac{x}{2} + c

) is correct because it is the result of integrating the simplified trigonometric series.

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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 Medium

376 of 411

Analyzing +4 | -1

\int e^{\sin x} \frac{\left(x \cos ^3 x-\sin x\right)}{\cos ^2 x} d x

is equal to

View answer and explanation

Correct option: C

e^{\sin x}(x-\sec x)+c

Option B (

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

) is correct because the integral can be expressed in the form

\int e^x [f(x) + f'(x)] dx

, where

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

and its derivative is

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

377 Medium

377 of 411

Application +4 | -1

\int \frac{1}{x \sqrt{1+\log x}} d x=

View answer and explanation

Correct option: C

2 \sqrt{1+\log x}+c

Option C is correct because the integral can be solved using substitution.

378 Medium

378 of 411

Application +4 | -1

\int \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\sqrt {{x^2} - \,\,1} }}\;\,\,dx

is equal to

View answer and explanation

Correct option: D

- \,\,\frac{{\sqrt {{x^2} - \,\,1} }}{{{x^2}}}\, + \,c

Option D (

\ - \,\,\frac{{\sqrt {{{{x^2} - \,\,1}}} }}{{{{x^2}}}}\, + \,\,c

) is correct because after trigonometric substitution and simplification, the result matches this option.

379 Medium

379 of 411

Application +4 | -1

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

View answer and explanation

Correct option: A

x-\log \left[1+\sqrt{1-e^{2 x}}\right]+c

Option A is correct because the integral can be solved by rewriting the integrand and using substitution, leading to the expression

x - \log[1 + \sqrt{1 - e^{2x}}]+c

380 Medium

380 of 411

Application +4 | -1

\int \frac{{{{\left( {{x^4} - \,\,x} \right)}^{1/4}}}}{{{x^5}}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{4}{{15}}\,\,{\left( {1 - \,\,\frac{1}{{{x^3}}}} \right)^{5/4}} + \,\,C

Option A is correct because the integral simplifies to a form where the power rule can be applied after a suitable substitution.

381 Medium

381 of 411

Application +4 | -1

\int e^{2 x} \frac{1+\sin 2 x}{1+\cos 2 x} d x=

View answer and explanation

Correct option: C

\frac{e^{2 x} \tan x}{2}+c

Option C is correct because the integral simplifies to

\frac{1}{2} \int e^{2x} \sec^2 x dx + \int e^{2x} \tan x dx

, which integrates to

\frac{e^{2x} \tan x}{2} + c

382 Medium

382 of 411

Understanding +4 | -1

\int \,\,\frac{{\cos \,\,x\,\, + \,\,x\,\,\sin \,\,x}}{{x\,\,\left( {x\,\, + \,\,\cos \,\,x} \right)}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\log \,\,\left( {\frac{x}{{x\,\, + \,\,\cos \,\,x}}} \right) + C

Option B (

\log \,\,\left( {\frac{x}{{x\,\, + \,\,\cos \,\,x}}} \right) + C

) is correct because the integral can be simplified and solved using basic integration rules.

383 Medium

383 of 411

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{1}{2}\left(\frac{e^x}{1+e^{2 x}}\right)+\frac{1}{2} \operatorname{Tan}^{-1} e^x+c

Option C (

\frac{1}{2}\left(\frac{e^x}{1+e^{2 x}}\right)+\frac{1}{2} \operatorname{Tan}^{-1} e^x+c

) is correct because by substituting

t=e^x

and then

t=\tan \theta

, we can evaluate the integral.

384 Medium

384 of 411

Application +4 | -1

\int \,\,\frac{1}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{{3\pi }}{8}} \right)\,} \right| + C

Option A (

\frac{1}{{\sqrt 2 }}\,\log \,\left| {\,\tan \,\left( {\frac{x}{2}\, + \\,\frac{{3\pi }}{8}} \right)\} \right| + C

) is correct because it is the result of integrating the given expression using trigonometric identities and substitution.

385 Medium

385 of 411

Application +4 | -1

I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x) d x

and

I(0)=

1 , then

I\left(\frac{\pi}{3}\right)

is equal to

View answer and explanation

Correct option: B

\frac{1}{2} e^{\frac{3}{4}}

Option B (

\frac{1}{2} e^{\frac{3}{4}}

) is correct because it involves solving the given integral and applying the initial condition.

386 Medium

386 of 411

Application +4 | -1

k\,\int \nolimits_0^1 \,\,x\,.\,f\left( {3x} \right)dx = \int \nolimits_0^3 \,t\,.\,f\,\,\left( t \right)\,\,dt

then the value of

k

View answer and explanation

Correct option: A

Option A (9) is correct because by substituting

t = 3x

, the integral transforms, and solving for

k

yields 9.

387 Medium

387 of 411

Application +4 | -1

\int \frac{d x}{\tan x+\cot x+\sec x+\operatorname{cosec} x}=

View answer and explanation

Correct option: B

\frac{1}{2}(\sin x-\cos x-x)+c

Option B is correct because the integral simplifies to an expression involving sine, cosine, and a constant, which can be directly integrated.

388 Medium

388 of 411

Analyzing +4 | -1

\int \frac{d x}{\cos ^6 x+\sin ^6 x}

equals

View answer and explanation

Correct option: B

\frac{1}{\sqrt{3}} \operatorname{Tan}^{-1} \frac{(\tan x-\cot x)}{\sqrt{3}}+c

Option B is correct because the integral simplifies to an arctangent form after trigonometric manipulation and substitution.

389 Medium

389 of 411

Application +4 | -1

Let

\int \frac{x^{1 / 2}}{\sqrt{1-x^3}} d x=\frac{2}{3} \operatorname{gof}(x)+c

then

View answer and explanation

Correct option: D

f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x

Option D (

f(x)=x^{3 / 2}, g(x)=\sin ^{-1} x

) is correct because it is the result of u-substitution and recognizing the form of the arcsine integral.

390 Medium

390 of 411

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

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

Option C (

e^x\left(\frac{{x+1}}{{x+2}}\right)+c

) is correct because the integral can be expressed in the form

\int e^x [f(x) + f'(x)] dx

, where

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

and its derivative is

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