$\smallint \,\,\cos \,\,\left[ {2\,\,{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} } \right]\,\,dx$ is equal to

$\frac{1}{2}\,\,\sin \,\,\left[ {2\,\,{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} } \right]\,\, + \,\,c$

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

$\mathop \smallint \nolimits_0^3 \,\,\frac{{3x\,\, + \,\,1}}{{{x^2}\,\, + \,\,9\;}}\;\,\,dx$ is equal to

$\log \,\,\left( {2\,\sqrt 2 } \right)\,\, + \,\,\frac{\pi }{2}$

$\log \,\,\left( {2\,\,\sqrt 2 } \right)\,\, + \,\,\frac{\pi }{6}$

$\log \,\,\left( {2\,\,\sqrt 2 } \right)\,\, + \,\,\frac{\pi }{3}$

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

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

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

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

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

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

If $I = \smallint \,\,\frac{1}{{{x^4}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} }}\,\,dx$, then I equals

$\frac{1}{{{a^4}}}\,\,\left\{ {\frac{1}{x}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} \,\, - \,\,\frac{1}{{3{x^3}}}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} } \right\} + C$

$\frac{1}{{{a^4}}}\,\,\left\{ {\frac{1}{x}\sqrt {{a^2}\,\, + \,\,{x^2}} - \,\,\frac{1}{{2\,\sqrt x }}{{\left( {{a^2}\,\, + \,\,{x^2}} \right)}^{3/2}}} \right\} + C$

If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to

$\frac{1}{2}\left[x^2\{F(x)\}^2-\int\{F(x)\}^2 d x\right]$

$\frac{1}{2}\left[x^2 F(x)-\frac{1}{2} \int\{F(x)\}^2 d x\right]$

$\smallint \,\,\frac{{\sin \,\,x}}{{\sin \,(x\,\, - \,\,\alpha )}}\,\,dx$ is

$x\,\,\sin \,\,\alpha \,\, + \,\,\cos \,\,\alpha \,\,\log \,\,\sin \,\,(x + \alpha )\,\, + \,\,c$

$x\,\,\sin \,\,\alpha \,\, + \,\,\cos \,\,\alpha \,\,\log \,\,\sin \,\,(x - \alpha )\,\, + \,\,c$

$x\,\,\cos \,\,\alpha \,\, + \,\,\sin \,\,\alpha \,\,\log \,\,\cos \,\,(x + \alpha )\,\, + \,\,c$

$\mathop \smallint \nolimits_0^\lambda \,\,\frac{{y\,\,dy}}{{\sqrt {y + \lambda } }}$ is equal to

$\frac{1}{3}\,\,\left( {2 - \sqrt 2 } \right)\,\,\lambda \,\,\sqrt \lambda $

$\frac{1}{3}\,\,\left( {2 + \sqrt 2 } \right)\,\,\lambda \,\,\sqrt \lambda $

Back to INDEFINITE INTEGRATION

Your Progress

0 / 995 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 721-750 of 995 questions

Preparing practice and saved progress…

721 Medium

721 of 995

Analyzing +4 | -1

f(x)

is a quadratic expression such that

f(0)=f(1)=3 f(2)=-3

, then

\int \frac{f(x)}{x^3-1} d x=

View answer and explanation

Correct option: A

\log _e|x-1|+\log _e\left(x^2+x+1\right)+\frac{2}{\sqrt{3}} \operatorname{Tan}^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+c

The correct answer is A (

\log _e|x-1|+\log _e(x^2+x+1)+\frac{2}{\sqrt{3}} \operatorname{Tan}^{-1}(\frac{2 x+1}{\sqrt{3}})+c

) because after determining the quadratic expression

f(x)

, the integral is solved using partial fraction decomposition and standard integration techniques.

722 Medium

722 of 995

Analyzing +4 | -1

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

View answer and explanation

Correct option: C

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

Option B (

\frac{{x e^x}}{{\sqrt{{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}}{{\sqrt{{x^2+1}}}}

and its derivative is

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

723 Easy

723 of 995

Application +4 | -1

\int \,\,\cos \,\,\left[ {2\,\,{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} } \right]\,\,dx

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the expression inside the cosine simplifies to -x, and the integral of -x is -x²/2 + c.

724 Easy

724 of 995

Analyzing +4 | -1

\int \nolimits_0^2 \,\,\frac{{2x\,\, - \,\,2}}{{2x\,\, - \,\,{x^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because substituting

t = 2x - x^2

makes the limits of integration the same, resulting in a zero integral.

725 Medium

725 of 995

Application +4 | -1

\int \nolimits_0^3 \,\,\frac{{3x\,\, + \,\,1}}{{{x^2}\,\, + \,\,9\;}}\;\,\,dx

is equal to

View answer and explanation

Correct option: A

\log \,\,(2\,\sqrt 2 )\,\, + \,\,\frac{\pi }{{12}}

Option A (

\log \,(2\sqrt 2 )\,+ \frac{\pi }{{12}}

) is correct because it results from correctly evaluating the definite integral after splitting it into two simpler integrals.

726 Medium

726 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the integral is solved by dividing by

x^2

and then using a substitution.

727 Easy

727 of 995

Application +4 | -1

\int \frac{1+\tan x}{x+\log \sec x} d x=

View answer and explanation

Correct option: A

\log (x+\log \sec x)+c

Option A is correct because the integral simplifies to

\log(x + \log \sec x) + c

by recognizing the numerator as the derivative of the denominator.

728 Easy

728 of 995

Application +4 | -1

f(x)=\frac{\log x}{x^3}

, then its antiderivative

F(x)

given by

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the result of integrating the given function using integration by parts.

729 Easy

729 of 995

Understanding +4 | -1

n

is an odd positive integer, then

\int\left|x^n\right| d x

is equal to

View answer and explanation

Correct option: C

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

Option C is correct because, for odd n, the integral of |x^n| dx results in an expression that involves |x|^n * x / (n+1) + C, accounting for both positive and negative values of x.

730 Easy

730 of 995

Application +4 | -1

\int \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3} x d x=

View answer and explanation

Correct option: B

-3(\tan x)^{-1 / 3}+c

Option B is correct because the integral can be simplified using trigonometric identities and solved using substitution, resulting in

-3(\tan x)^{-1 / 3}+c

731 Easy

731 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B,

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

, is correct because dividing by

\cos^2 x

and using a u-substitution leads to this result.

732 Easy

732 of 995

Analyzing +4 | -1

\int \sqrt{x-3}\left\{\sin ^{-1}(\ln x)+\cos ^{-1}(\ln x)\right\} d x

is equal to

View answer and explanation

Correct option: C

Does not exist

The correct answer is D (Does not exist) because the domain of

\sin^{-1}(\ln x) + \cos^{-1}(\ln x)

is restricted to

-1 \leq \ln x \leq 1

, which is not always true.

733 Easy

733 of 995

Application +4 | -1

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

is equal to

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.

734 Medium

734 of 995

Analyzing +4 | -1

I = \int \,\,\frac{1}{{{x^4}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} }}\,\,dx

, then I equals

View answer and explanation

Correct option: B

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

Option D (

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

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

735 Medium

735 of 995

Application +4 | -1

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

, then

f(x)

is equal to

View answer and explanation

Correct option: C

\tan \frac{x}{2}

Option C (

\tan \frac{x}{2}

) is correct because it is the function that satisfies the given integral equation after simplification and integration.

Explore Other Chapters in MATHEMATICS

View All →

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

736 Medium

736 of 995

Application +4 | -1

\int f\left( x \right)\,\,\sin \,\,x\,\,\cos \,\,x\,\,\;dx = \frac{1}{{2\,\,\left( {{b^2} - \,\,{a^2}} \right)}}\log \,\,\left\{ {f\left( x \right)} \right\} + C

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: A

\frac{1}{{{a^2}\,\,{{\sin }^2}x\,\, + \,\,{b^2}\,\,{{\cos }^2}\,\,x}}

Option A (

\frac{1}{{{a^2}\,{{\sin }^2}x\, + \,{b^2}\,{{\cos }^2}\,x}}

) is correct because by differentiating the given equation and solving for

f(x)

, we find that

f(x)

matches this form.

737 Easy

737 of 995

Application +4 | -1

\int \,\,\frac{{1\,\, + \,\,\tan \,\,x}}{{{e^{ - x}}\,\,\cos \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the integral simplifies to this expression after applying integration by parts.

738 Easy

738 of 995

Application +4 | -1

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

, then

\int x^3 f\left(x^2\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]

Option B is correct. The integral is solved using substitution and integration by parts, resulting in the given expression.

739 Easy

739 of 995

Application +4 | -1

\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right)} \,{e^{({e^x} + {e^{ - x}})}}dx = g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c,

where

c

is a constant of integration, then

g\left( 0 \right)

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because after solving the integral and substituting x=0 into the resulting function g(x), we obtain the value 2.

740 Medium

740 of 995

Application +4 | -1

\int \frac{2^{x+1}-5^{x-1}}{10^x} d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of integrating the given expression using exponential integral formulas.

741 Easy

741 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A (

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

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

742 Medium

742 of 995

Application +4 | -1

\int \,\,\frac{{\sin \,\,x}}{{\sin \,(x\,\, - \,\,\alpha )}}\,\,dx

View answer and explanation

Correct option: D

x\,\,\cos \,\,\alpha \,\, + \,\,\sin \,\,\alpha \,\,\log \,\,\sin \,\,\left( {x - \alpha } \right)\,\, + \,\,c

Option D (

x\cos \alpha + \sin \alpha \log \sin (x - \alpha ) + c

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

743 Easy

743 of 995

Application +4 | -1

\int e^x\left(\frac{x+2}{x+4}\right)^2 d x=

View answer and explanation

Correct option: A

\frac{x e^x}{x+4}+c

Option A (

\frac{x e^x}{x+4}+c

) is correct because the integral is solved using the form

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

744 Medium

744 of 995

Application +4 | -1

\int \nolimits_0^\lambda \,\,\frac{{y\,\,dy}}{{\sqrt {y + \lambda } }}

is equal to

View answer and explanation

Correct option: A

\frac{2}{3}\,\,\left( {2 - \sqrt 2 } \right)\,\,\lambda \,\,\sqrt \lambda

Option A is correct because after performing the substitution and evaluating the definite integral, the result matches the given expression.

745 Easy

745 of 995

Application +4 | -1

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

View answer and explanation

Correct option: D

e^x \tan x+c

Option D (

e^x \tan x+c

) is correct because the integral simplifies to this expression by recognizing the form of

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

746 Easy

746 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

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

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

747 Easy

747 of 995

Application +4 | -1

\int e^x \sin x d x=\frac{1}{2} e^x \cdot a+c

, then

a=

View answer and explanation

Correct option: A

\sin x-\cos x

Option A (

\sin x-\cos x

) is correct because it is the result of solving the integral

\int e^x \sin x dx

using integration by parts twice and comparing with the given form.

748 Easy

748 of 995

Application +4 | -1

\int x^3 \log x d x=

View answer and explanation

Correct option: B

\frac{1}{16}\left[4 x^4 \log x-x^4\right]+c

Option B is correct because the integral of x³ log(x) dx is found using integration by parts, resulting in 1/16[4x⁴ log(x) - x⁴] + c.

749 Medium

749 of 995

Analyzing +4 | -1

\int x\,\,\log \,\,\left( {1\,\, + \,\,1/x} \right)\,\,dx = f\,\,\left( x \right)\,\, \cdot \,\,\log \,\,\left( {x\,\, + \,\,1} \right)\,\, + \,\,g\left( x \right)\,\, \cdot \,\,{x^2}\,\, + \,\,Ax + C

, then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given conditions do not match the derived functions f(x), g(x), and the value of A after integration and comparison.

750 Easy

750 of 995

Application +4 | -1

\int \sin 2 x \log _e \cos x d x

is equal to

View answer and explanation

Correct option: C

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

Option C,

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

, is correct because it results from integrating

\sin 2x \log_e \cos x

using integration by parts and substitution.

Revision notes and key takeaways

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

Showing 721-750 of 995 questions

Your Progress

Indefinite integration practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice