$\smallint \frac{{dx}}{{x\,\left( {{x^n} + 1} \right)}}$ is equal to

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

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

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

$\frac{1}{4}\,\,\log \,\,\left( {1 + \sqrt x \,} \right) + c$

$3\,\,\log \,\,\left( {1 + \sqrt x \,} \right) + c$

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

$\frac{{{{\sin }^5}\,\,x}}{5}\,\, - \,\,\frac{{{{\sin }^3}\,\,x}}{3}\,\, + \,\,c$

$\frac{{{{\sin }^5}\,\,x}}{5}\,\, + \,\,\frac{{{{\sin }^3}\,\,x}}{3}\,\, + \,\,c$

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

$\frac{{\sqrt {{x^2}\,\, + \,\,2x\,\, + \,\,2} }}{{{{\left( {x\,\, + \,\,1} \right)}^2}}}\,\, + \,\,C$

$\smallint \sqrt {1\,\, + \,\,\cos \,\,x} \,dx$ is equal to

$2\,\,\sqrt 2 \,\,\cos \,\,\frac{x}{2}\,\, + \,\,c$

$\sqrt 2 \,\,\cos \,\,\frac{x}{2}\,\, + \,\,c$

$\smallint \,\,\frac{{1\,\, + \,\,x\,\, + \,\,\sqrt {x\,\, + \,\,{x^2}} }}{{\sqrt x \,\, + \,\,\sqrt {1\,\, + \,\,x} }}\,\,dx = $

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

$\sqrt {1\,\, + \,\,x} \,\, + \,\,C$

$2\,\,{\left( {1\,\, + \,\,x} \right)^{3/2}} + \,\,C$

$\smallint \,\,\frac{{d\,(\cos \,\,{\rm{\theta }})}}{{\sqrt {1\,\, - \,\,{{\cos }^2}\,{\rm{\theta }}} }}$ is equal to

${\sin ^{ - 1}}\,\,(\cos \,\,{\rm{\theta }})\,\, + \,\,c$

${\cos ^{ - 1}}\,\,{\rm{\theta }}\,\, + \,\,c$

${\sin ^{ - 1}}\,\,{\rm{\theta }}\,\, + \,\,c$

$\int \frac{d x}{\sqrt[4]{1+x^4}}$ is

$\frac{1}{4} \log _e\left|\frac{(1+x)^{1 / 4}-x}{(1+x)^{1 / 4}+x}\right|-\frac{1}{2} \operatorname{Tan}^{-1}\left(\frac{(1+x)^{1 / 4}}{x}\right)+c$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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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 601-630 of 995 questions

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601 Medium

601 of 995

Application +4 | -1

The value of

\int \,\,\frac{1}{{{{\left[ {{{\left( {x - 1} \right)}^3}\,\,{{\left( {x + 2} \right)}^5}} \right]}^{1/4}}}}\,\,\;dx

View answer and explanation

Correct option: A

\frac{4}{3}\,\,{\left( {\frac{{x - 1}}{{x + 2}}} \right)^{1/4}} + \,\,c

Option A (

\frac{4}{3}\left( {\frac{{x - 1}}{{x + 2}}} \right)^{1/4} + c

) is correct because it is the result of evaluating the indefinite integral using substitution.

602 Hard

602 of 995

Understanding +4 | -1

\int\left(\frac{f(x) g^{\prime}(x)-f^{\prime}(x) g(x)}{f(x) g(x)}\right)(\log (g(x))-\log (f(x))) d x

is equal to:

View answer and explanation

Correct option: C

\frac{1}{2}\left(\log \left(\frac{g(x)}{f(x)}\right)\right)^2+C

Option C (

\frac{{1}}{{2}}\left(\log \left(\frac{{g(x)}}{{f(x)}}\right)\right)^2+C

) is correct because it is the result of integrating the given expression after a suitable substitution.

603 Easy

603 of 995

Analyzing +4 | -1

\int \frac{1}{\sin ^6 x+\cos ^6 x} d x

is equal to

View answer and explanation

Correct option: C

\tan ^{-1}(\tan x-\cot x)+C

Option C (

\tan ^{-1}(\tan x-\cot x)+C

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

604 Easy

604 of 995

Application +4 | -1

\int \cos ^{-1}\left(\frac{1}{x}\right) d x

View answer and explanation

Correct option: B

x \sec ^{-1} x-\cosh ^{-1} x+C

Option B is correct because the integral of arccos(1/x) dx is equivalent to the integral of arcsec(x) dx, which after integration by parts results in x arcsec(x) - arccosh(x) + C.

605 Medium

605 of 995

Application +4 | -1

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

, then

I_n+n I_{n-1}

is equal to

View answer and explanation

Correct option: B

x(\ln x)^n

Option B (

x(\ln x)^n

) is correct because it represents the result of integration by parts applied to the given integral.

606 Easy

606 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A,

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

, is correct because it is the result of integrating the given function using substitution and partial fractions.

607 Easy

607 of 995

Application +4 | -1

\int \left( {\sin \,\,2x - \cos \,\,\,2x} \right)dx = \frac{1}{{\sqrt 2 }}\,\,\sin \,\,\left( {2x - a} \right) + b

, then

View answer and explanation

Correct option: B

a = - \frac{{5\pi }}{4},\,\;b \in R

Option B (

a = - \frac{{5\pi }}{4},\;b \in R

) is correct because it is derived by integrating the given trigonometric expression and comparing it with the target form.

608 Easy

608 of 995

Analyzing +4 | -1

\int \sqrt{\frac{x}{a^3-x^3}} d x=m \sin ^{-1}\left(\frac{x}{a}\right)^n+C

, then

View answer and explanation

Correct option: C

m=1 / n

Option C is correct because after evaluating the integral, we find that

m = \frac{2}{3}

and

n = \frac{3}{2}

, which means

m = \frac{1}{n}

609 Easy

609 of 995

Application +4 | -1

\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x

equals to (for some arbitrary constant

K

)

View answer and explanation

Correct option: C

\frac{-1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K

Option C (

\frac{{-1}}{{(\sec x+\tan x)^{{11 / 2}}}}\left{\frac{{1}}{{11}}+\frac{{1}}{{7}}(\sec x+\tan x)^2\right\}+K

) is correct because it represents the result of the given integral after applying substitution and simplification.

610 Easy

610 of 995

Application +4 | -1

\int x^n \log x d x=

View answer and explanation

Correct option: D

\frac{x^{n+1}}{n+1}\left\{\log x-\frac{1}{n+1}\right\}+c

Option D (

\frac{x^{n+1}}{n+1}\left{\log x-\frac{1}{n+1}\right}+c

) is correct because it is the result of integrating

x^n \log x

using integration by parts.

611 Easy

611 of 995

Application +4 | -1

\int \,\,\frac{{dx}}{{x + \sqrt x }}

is equal to

View answer and explanation

Correct option: B

2\,\,\log \,\,\left( {1 + \sqrt x \,} \right)\, + c

Option B (

2\,\log \,\left( {1 + \sqrt x \} \right)\, + c

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

612 Easy

612 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

\sqrt{1+x^2}+\log \left\{x+\sqrt{1+x^2}\right\}+c

Option C is correct because the integral can be split into two parts, each solved by a different method.

613 Medium

613 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{{{{\cos }^5}\,\,x}}{5}\,\, - \,\,\frac{{{{\cos }^3}\,\,x}}{3}\,\, + \,\,c

Option C (

\frac{{{{\cos }^5}\,\,x}}{5}\,\, - \,\,\frac{{{{\cos }^3}\,\,x}}{3}\,\, + \,\,c

) is correct because it results from correctly applying u-substitution with

u = \cos x

to the given integral.

614 Easy

614 of 995

Application +4 | -1

\int \,\,\frac{{\cos \,\,2x}}{{\cos \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

2\,\,\sin \,\,x + \log \,\,\left( {\sec \,\,x - \,\tan x} \right) + C

Option A (

2\,\sin \,x + \log \,\left( {\sec \,x - \,\tan x} \right) + C

) is correct because after simplifying the integral using trigonometric identities and integrating, the result is

2\sin x + \log(\sec x - \tan x) + C

615 Easy

615 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

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

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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

616 Medium

616 of 995

Application +4 | -1

Evaluate:

\int \sin ^3 x \cos ^3 x d x

View answer and explanation

Correct option: B

\frac{1}{32}\left\{-\frac{3}{2} \cos 2 x+\frac{1}{6} \cos 6 x\right\}+C

Option B (

\frac{1}{32}\{{-\frac{3}{2} \cos 2 x+\frac{1}{6} \cos 6 x\}}+C

) is correct because it is the result of integrating the given trigonometric expression after simplification.

617 Easy

617 of 995

Application +4 | -1

\int \frac{d x}{1+3 \sin ^2 x}=

View answer and explanation

Correct option: B

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

Option B is correct because the integral is solved by dividing by cos²x and using a substitution, resulting in

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

618 Easy

618 of 995

Application +4 | -1

\int \frac{1}{7+5 \cos x} d x=

View answer and explanation

Correct option: A

\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+C

Option A,

\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+C

, is correct because using the half-angle substitution and simplifying leads to this result.

619 Easy

619 of 995

Application +4 | -1

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

and

f(0)=0

, then

f(1)

equals:

View answer and explanation

Correct option: C

\tan 1+\frac{\pi}{4}

Option C (

\tan 1+\frac{\pi}{4}

) is correct because evaluating the integral and applying the initial condition leads to the given result.

620 Easy

620 of 995

Application +4 | -1

For any natural number

m

\int\left(x^{7 m}+x^{2 m}+x^m\right)\left(2 x^{6 m}+7 x^m+14\right)^{1 / m} d x

, where

x>0

equals

View answer and explanation

Correct option: C

\frac{\left(2 x^{7 m}+7 x^{2 m}+14 x^m\right)^{\frac{m+1}{m}}}{14(m+1)}+C

Option C is correct because the integral can be solved by substitution, leading to the expression

\frac{\left(2 x^{7 m}+7 x^{2 m}+14 x^m\right)^{\frac{m+1}{m}}}{14(m+1)}+C

621 Easy

621 of 995

Application +4 | -1

\int \sqrt {1\,\, + \,\,\cos \,\,x} \,dx

is equal to

View answer and explanation

Correct option: B

2\,\,\sqrt 2 \,\,\sin \,\,\frac{x}{2}\,\, + \,\,c

Option B,

2\,\,\sqrt 2 \,\,\sin \,\,\frac{x}{2}\,\, + \,\,c

, is correct because using trigonometric identities, the integral can be simplified and solved, resulting in this expression.

622 Easy

622 of 995

Application +4 | -1

\int(x+1)^2 e^x d x

is equal to

View answer and explanation

Correct option: D

\left(x^2+1\right) e^x+C

Option D (

\left(x^2+1\right) e^x+C

) is correct because after expanding and integrating by parts, the integral simplifies to this expression.

623 Medium

623 of 995

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

\frac{2}{3}\,\,{\left( {1\,\, + \,\,x} \right)^{3/2}}\,\, + \,\,C

Option B (

\frac{2}{3}{\left( {1 + x} \right)^{3/2}} + C

) is correct because the integral simplifies to

\int \sqrt{1+x} dx

, which integrates to the given expression.

624 Medium

624 of 995

Understanding +4 | -1

\int \,\,\frac{{d\,(\cos \,\,\mathrm{\theta })}}{{\sqrt {1\,\, - \,\,{{\cos }^2}\,\mathrm{\theta }} }}

is equal to

View answer and explanation

Correct option: D

{\sin ^{ - 1}}\,\,(\cos \,\,\mathrm{\theta })\,\, + \,\,c

Option D (

{\sin ^{ - 1}}\,(\cos \,\mathrm{\theta })\, + \,c

) is correct because the integral can be directly computed using the definition of arcsin.

625 Medium

625 of 995

Analyzing +4 | -1

\int \frac{d x}{\sqrt[4]{1+x^4}}

View answer and explanation

Correct option: C

\frac{-1}{4} \log _e\left|\frac{\left(1+x^4\right)^{1 / 4}-x}{\left(1+x^4\right)^{1 / 4}+x}\right|-\frac{1}{2} \operatorname{Tan}^{-1}\left(\frac{\left(1+x^4\right)^{1 / 4}}{x}\right)+c

Option C (

\frac{{-1}}{{4}} \log _e|\frac{{(1+x^4)^{1 / 4}-x}}{{(1+x^4)^{1 / 4}+x}}|-\frac{{1}}{{2}} \operatorname{{Tan}}^{-1}(\frac{{(1+x^4)^{1 / 4}}}{{x}})+c

) is correct because after applying the appropriate substitution and simplifying, the integral evaluates to this form.

626 Easy

626 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C is correct because the integral is solved by recognizing the numerator as half the derivative of the denominator, leading to a logarithmic solution.

627 Easy

627 of 995

Analyzing +4 | -1

The value of

I=\int \frac{d x}{2 x \sqrt{1-x} \sqrt{(2-x)+\sqrt{1-x}}}

=-\frac{1}{2}\left\{\log \left(z+\frac{3}{2}+\sqrt{z^2+3 z+3}\right)\right\}+\frac{1}{2}\left|\log s-\frac{1}{2}+\sqrt{s^2-s+1}\right|+C

and

s-z=\frac{k}{x}

, then value of

k

, is

View answer and explanation

Correct option: B

Option B is correct because after performing the integration and substituting back, the value of k in s-z = k/x is found to be 2.

628 Easy

628 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D is correct because the integral simplifies to

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

after a suitable substitution.

629 Medium

629 of 995

Application +4 | -1

\int \frac{d x}{(2 x+3) \sqrt{4 x+5}}

is equal to

View answer and explanation

Correct option: B

\tan ^{-1} \sqrt{4 x+5}+c

Option B (

\tan ^{-1} \sqrt{4 x+5}+c

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

630 Easy

630 of 995

Application +4 | -1

\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x-\cos x}{b}\right)+C

, then

(a, b)

is equal to

View answer and explanation

Correct option: A

(1,3)

Option A,

(1,3)

, is correct because manipulating the integrand and completing the square leads to this result.

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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