$\smallint \,\frac{{a{x^3} + \,\,b{x^2} + c}}{{{x^4}}}\;\,dx$ equals

$a\,\,\log \,\,x\,\, + \,\,\frac{b}{{{x^2}}}\,\, + \,\,\frac{c}{{3{x^3}}}\,\, + \,\,k$

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

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

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

$\frac{1}{8}\,\,\left( {\sin \,\,\frac{x}{8}\,\, - \,\,\cos \,\,\frac{x}{8}} \right) + c$

$8\,\,\left( {\cos \,\,\frac{x}{8}\,\, - \,\,\sin \,\,\frac{x}{8}} \right) + c$

But for all arbitrary constants, $\smallint \,\,\sqrt {\frac{{1\,\, + \,\,\sin \,\,\theta \,\, - \,\,{{\sin }^2}\,\,\theta \,\, - \,\,{{\sin }^3}\,\,\theta }}{{2\,\,\sin \,\,\theta \,\, - \,\,1}}\,\,} d\theta \,\,{\rm{is }}\,{\rm{equal }}\,{\rm{to}}$

$\frac{1}{{2\,\,\sqrt 2 }}\,\,\sqrt {\sin \,\,\theta \,\, - \,\,\cos \,\,2\theta } \,\, + \,\,\frac{3}{4}\,\,{\log _e}\,\,\left| {\,\left( {4\,\,\sin \,\,\theta \,\, + \,\,1} \right)\,\, - \,\,\sqrt {\sin \,\,\theta \,\, - \,\,\cos \,\,2\,\,\theta } \,} \

$\frac{1}{2}\,\,\sqrt {\,\,\sin \,\,\theta \,\, + \,\,\cos \,\,2\,\theta } \,\, + \,\,\frac{3}{{4\,\,\sqrt 2 }}\,\,{\log _e}\,\,\left| {\,4\,\,\sin \,\,\theta \,\, + \,\,1\,\, - \,\,\sqrt {\sin \,\,\theta \,\, - \,\,\cos \,\,2\,\,\theta \,} } \right|$

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

$\sqrt {\tan \,\,x} \,\,\, + \,\,\frac{2}{5}\,\,{\tan ^{5/2}}x\,\, + \,\,c$

$2\,\sqrt {\tan \,\,x} \,\, + \,\,\frac{2}{5}\,\,{\tan ^{5/2}}x\,\, + \,\,c$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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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 1-30 of 995 questions

1 Easy

1 of 995

Application +4 | -1

\int \sin ^5 x \cos ^4 x d x=

View answer and explanation

Correct option: A

-\frac{1}{5} \cos ^5 x+\frac{2}{7} \cos ^7 x-\frac{1}{9} \cos ^9 x+c

Option A is correct because the integral is solved by substituting

\cos x = t

and simplifying the resulting expression.

2 Easy

2 of 995

Analyzing +4 | -1

\int \frac{1-7 \cos ^2 x}{\sin ^7 x \cos ^2 x} d x=\frac{f(x)}{\sin ^7 x}+C

is equal to

View answer and explanation

Correct option: C

\tan x

Option D (

\cot x

) is correct because by rewriting the integrand and integrating,

f(x)

is found to be

\tan x

, which is equivalent to

\frac{\sin x}{\cos x}

and simplifies to

\tan x

3 Easy

3 of 995

Application +4 | -1

\int \,\frac{{a{x^3} + \,\,b{x^2} + c}}{{{x^4}}}\;\,dx

equals

View answer and explanation

Correct option: C

a\,\,\log \,\,x\,\, - \,\,\frac{b}{x}\,\, - \,\,\frac{c}{{3{x^3}}}\,\, + \,\,k

Option C,

a\,\,\log \,x\,\, - \,\,\frac{b}{x}\,\, - \,\,\frac{c}{{3{x^3}}}\,\, + \,\,k

, is correct because it is the result of directly integrating each term after dividing by

x^4

4 Easy

4 of 995

Application +4 | -1

\int \frac{x^2+2}{x^4+4} d x

equals

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because after algebraic manipulation and substitution, the integral evaluates to this expression.

5 Easy

5 of 995

Application +4 | -1

\int {\sin ^{ - 1}}\,\,x\,\,dx

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of integrating arcsin(x) by parts.

6 Medium

6 of 995

Application +4 | -1

The value of

\int \limits_0^{\pi /2} \,\,\frac{{{{\sin }^3}\,\,x\,\,\cos \,\,x}}{{{{\sin }^4}\,\,x\,\, + \,\,{{\cos }^4}\,\,x}}\,\,dx

, is

View answer and explanation

Correct option: A

\pi /8

Option A (π/8) is correct because by using the property of definite integrals and trigonometric identities, the integral simplifies to π/8.

7 Easy

7 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{1}{3} \log \left(e^x-1\right)-\frac{1}{3} \log \left(e^x+2\right)+c

Option A is correct because the integral is solved by multiplying the numerator and denominator by

e^x

, then substituting

t = e^x

, and using partial fraction decomposition.

8 Easy

8 of 995

Application +4 | -1

\int x \sin x \sec ^3 x d x=

View answer and explanation

Correct option: B

\frac{1}{2}\left[x \sec ^2 x-\tan x\right]+c

Option B (

\frac{{1}}{{2}}\left[x \sec ^2 x-\tan x\right]+c

) is correct because it is the result of integrating the given expression using substitution and integration by parts.

9 Easy

9 of 995

Application +4 | -1

\int \sqrt{\frac{x^4}{a^6+x^6}} d x=g(x)+C

, then

g(x)=

View answer and explanation

Correct option: C

\frac{1}{3} \log \left|x^3+\sqrt{a^6+x^6}\right|

Option C,

\frac{1}{3} \log \left|x^3+\sqrt{a^6+x^6}\right|

, is the correct antiderivative obtained by substitution and using the integral formula for

\int \frac{1}{\sqrt{u^2+a^2}} du

10 Easy

10 of 995

Application +4 | -1

\int \cos ^5 x d x=

View answer and explanation

Correct option: A

\sin x-\frac{2}{3} \sin ^3 x+\frac{1}{5} \sin ^5 x+c

Option A is correct because the integral can be solved by rewriting the integrand and using substitution, resulting in

\sin x-\frac{2}{3} \sin ^3 x+\frac{1}{5} \sin ^5 x+c

11 Easy

11 of 995

Application +4 | -1

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

is equal

View answer and explanation

Correct option: B

\frac{1}{4} \log \left(x^4+1\right)+C

Option B (

\frac{1}{4} \log \left(x^4+1\right)+C

) is correct because after simplifying the integral, the result matches this option.

12 Easy

12 of 995

Analyzing +4 | -1

The integral

\int {\frac{{3{x^{13}} + 2{x^{11}}}}{{{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}dx}

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

View answer and explanation

Correct option: B

\frac{{{x^{12}}}}{{6{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C

Option B (

\frac{{{x^{12}}}}{{6{{\left( {2{x^4} + 3{x^2} + 1} \right)}^3}}} + C

) is correct because after dividing the numerator and denominator by

{x^{16}}

and substituting, the integral can be easily evaluated.

13 Easy

13 of 995

Analyzing +4 | -1

The value of

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

, is equal to

View answer and explanation

Correct option: B

\log \left|\frac{x e^x}{1+x e^x}\right|+\frac{1}{1+x e^x}+C

Option B (

\log \left|\frac{x e^x}{1+x e^x}\right|+\frac{1}{1+x e^x}+C

) is correct because the integral simplifies to this logarithmic form after appropriate substitutions and partial fraction decomposition.

14 Easy

14 of 995

Application +4 | -1

\int \sqrt {1 + \sin \,\,\frac{x}{4}} \,dx

is equal to

View answer and explanation

Correct option: A

8\,\,\left( {\sin \,\,\frac{x}{8}\,\, - \,\,\cos \,\,\frac{x}{8}} \right) + c

Option A (

8\,\,\left( {\sin \,\,\frac{x}{8}\,\, - \,\,\cos \,\,\frac{x}{8}} \right) + c

) is correct because it is the result of simplifying the integrand using trigonometric identities and integrating.

15 Easy

15 of 995

Application +4 | -1

\int \frac{\sec x}{\sqrt{\sin (2 x+\alpha)+\sin \alpha}} d x

is equal to

View answer and explanation

Correct option: A

\sqrt{2 \sec \alpha(\tan x+\tan \alpha)}+C

Option A is correct because after applying trigonometric identities and simplifying, the integral evaluates to the given expression involving

\tan x

and

\tan \alpha

Explore Other Chapters in MATHEMATICS

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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

16 Easy

16 of 995

Analyzing +4 | -1

\int \frac{1-7 \cos ^2 x}{\sin ^7 x \cos ^2 x} d x=\frac{f(x)}{(\sin x)^7}+C

, then

f(x)

is equal to

View answer and explanation

Correct option: C

\tan x

Option C (

\tan x

) is correct because integrating the given expression results in a function of the form f(x)/(sin x)⁷ + C, where f(x) = tan x.

17 Easy

17 of 995

Application +4 | -1

The integral

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

is equal to

View answer and explanation

Correct option: C

x e^{\tan ^{-1} x}+C

Option C (

x e^{\tan ^{-1} x}+C

) is correct because the integral can be solved using substitution and the formula

\int e^x(f(x)+f'(x)) dx = e^x f(x) + C

18 Easy

18 of 995

Application +4 | -1

\int \frac{e^{m \tan ^{-1} x}}{1+x^2} d x

equals to

View answer and explanation

Correct option: C

\frac{1}{m} e^{m \tan ^{-1} x}

Option C (

\frac{{1}}{{m}} e^{{m \tan^{{-1}} x}}

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

19 Medium

19 of 995

Analyzing +4 | -1

y(x-y)^2=x

, then

\int \frac{d x}{(x-3 y)}=\frac{m}{n} \ln \left[(x-y)^2-1\right]

. Then

(m+2 n)

is equal to

View answer and explanation

Correct option: C

Option C is correct because after solving the differential equation and substituting back, the value of (m+2n) is found to be 5.

20 Easy

20 of 995

Application +4 | -1

The antiderivative of

\frac{{{3^x}}}{{\sqrt {1 - {9^x}} }}

with respect to

x

View answer and explanation

Correct option: A

\left( {{{\log }_3}\,\,e} \right)\,\,{\sin ^{ - 1}}\,\,\left( {{3^x}} \right) + C

Option A (

\left( {{{\log }_3}\,e} \right)\,{\sin ^{ - 1}}\,\left( {{3^x}} \right) + C

) is correct because it is the antiderivative obtained by using substitution and the formula for the integral of

\frac{1}{\sqrt{a^2 - x^2}}

21 Easy

21 of 995

Application +4 | -1

I_n=\int(\log x)^n d x

, then

I_n+n I_{n-1}=

View answer and explanation

Correct option: B

x(\log x)^n

Option B (

x(\log x)^n

) is correct because integration by parts simplifies the expression to the given result.

22 Easy

22 of 995

Application +4 | -1

The integral

I=\int \frac{e^{3 \log _e 2 x}+5 e^{2 \log _e 2 x}}{e^{4 \log _e x+5 e^{3 \log _e x}-7 e^{2 \log _e x}}} d x

is equal to

View answer and explanation

Correct option: B

4 \ln \left|x^2+5 x-7\right|+C

Option B (

4 \ln \left|x^2+5 x-7\right|+C

) is correct because after simplifying the integral, the result matches this option.

23 Easy

23 of 995

Application +4 | -1

\int \frac{1}{\sin ^4 x+\cos ^4 x} d x

equals

View answer and explanation

Correct option: A

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

Option A is correct because the integral simplifies to a form that can be solved using the arctangent function, resulting in the given expression.

24 Easy

24 of 995

Analyzing +4 | -1

Let

f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x \, (x \geq 0)

. Then,

f(3)-f(1)

is equal to

View answer and explanation

Correct option: C

\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}

Option C (

\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}

) is correct because after using the substitution

x = \tan^2 \theta

and evaluating the definite integral, we get the result in option C.

25 Hard

25 of 995

Analyzing +4 | -1

But for all arbitrary constants,

\int \,\,\sqrt {\frac{{1\,\, + \,\,\sin \,\,\theta \,\, - \,\,{{\sin }^2}\,\,\theta \,\, - \,\,{{\sin }^3}\,\,\theta }}{{2\,\,\sin \,\,\theta \,\, - \,\,1}}\,\,} d\theta \,\,\mathrm{is }\,\mathrm{equal }\,\mathrm{to}

View answer and explanation

Correct option: A

\frac{1}{2}\,\,\sqrt {\sin \,\,\theta \,\, - \,\,\cos \,\,2\,\theta } \,\, + \,\,\frac{3}{{4\,\,\sqrt 2 }}\,\,{\log _e}\left| {\,\left( {4\,\,\sin \,\,\theta \,\, + \,\,1} \right)\,\, + \,\,2\,\,\sqrt 2 \,\,\sqrt {\sin \,\,\theta \,\, - \,\,\cos \,\,2\,

Option A is correct because the integral simplifies to a form that can be solved using substitution and partial fractions, resulting in the given expression.

26 Easy

26 of 995

Application +4 | -1

\int \frac{\sec ^2 x d x}{\sqrt{\tan ^2 x+4}}=

View answer and explanation

Correct option: A

\log \left[\tan x+\sqrt{\tan ^2 x+4}\right]+c

Option A (

\log \left[\tan x+\sqrt{\tan ^2 x+4}\right]+c

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

\frac{1}{\sqrt{t^2+a^2}}

27 Easy

27 of 995

Analyzing +4 | -1

\int \frac{5 x+4}{\sqrt{x^2+2 x+5}} d x=\left(x^2+2 x+5\right)^{1 / 5}+f(x)+c

then

f (x)

is equal to

View answer and explanation

Correct option: B

-\log _e\left(x+1+\sqrt{x^2+2 x+5}\right)

The correct answer is B (

-\log _e(x+1+\sqrt{x^2+2 x+5})

) because it results from correctly evaluating the integral after completing the square and using a suitable substitution.

28 Easy

28 of 995

Analyzing +4 | -1

\int \,\,\frac{{dx}}{{{{\cos }^3}\,\,\sqrt {2\,\,\sin \,\,2x} }}

is equal to

View answer and explanation

Correct option: A

\sqrt {\tan \,\,x} \,\,\, + \,\,\frac{{{{\tan }^{5/2}}x}}{5}\,\, + \,\,c

Option A (

\sqrt {\tan \\,\\,x} \\,\\,\\ + \\,\frac{{{{\tan }^{5/2}}x}}{5}\\,\\ + \\,\\,c

) is correct because it is the result of the integration after appropriate trigonometric manipulation and substitution.

29 Easy

29 of 995

Analyzing +4 | -1

\int\left(\frac{\sin \theta}{\cos 3 \theta}+\frac{\sin 3 \theta}{\cos 9 \theta}+\frac{\sin 9 \theta}{\cos 27 \theta}\right) d \theta

is equal to

View answer and explanation

Correct option: C

\frac{1}{2} \log \left|\frac{\sqrt[27]{\sec 27 \theta}}{\sec \theta}\right|+C

Option C (

\frac{1}{2} \log \left|\frac{\sqrt[27]{\sec 27 \theta}}{\sec \theta}\right|+C

) is correct because after simplifying the trigonometric terms and integrating, the result matches this option.

30 Easy

30 of 995

Application +4 | -1

\int {{x^5}{e^{ - 4{x^3}}}dx = \frac{1}{{48}}{e^{ - 4{x^3}}}f\left( x \right) + C}

, where

C

is a constant of integration, then

f(x)

is equal to

View answer and explanation

Correct option: A

- 4{x^3} - 1

Option A (

- 4{x^3} - 1

) is correct because it is the function

f(x)

obtained after performing integration by parts and comparing with the given equation.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 995 single-correct-answer MCQs on INDEFINITE INTEGRATION.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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