$\int \frac{x^3-x^2+x-1}{x^5+1} d x$ is

$\frac{1}{5} \ln \left(\frac{1+x^5}{\left(1+x^3\right)^5}\right)+c$

$\frac{1}{5} \ln \left(\frac{1+x^5}{1+x^3}\right)+c$

$\frac{1}{5} \ln \left(\frac{1+x^5}{1+x}\right)+c$

$\int\left[x+{\sqrt{x^2+1}}^n d x \,(n \neq \pm 1)\right.$ is

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

$\frac{1}{2}\left(x+\sqrt{x^2+1}\right)^n\left[\frac{x+\sqrt{x^2+1}}{n+1}+\frac{1}{n}\right]+c$

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

$\smallint \,\,\left[ {f\left( x \right)\,\,{\rm{g''}}\,\,\left( x \right)\,\, - \,\,f''\,\,\left( x \right)\,\,{\rm{g}}\,\,\left( x \right)} \right]\,\,dx$ is equal to

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

$\frac{{f\,\,\left( x \right)}}{{{\rm{g'}}\,\,\left( x \right)}}$

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

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

$\smallint \,{e^{\sin {\rm{\theta }}}}\,[\log \,\,\sin \,\,{\rm{\theta }} + \,\,{\rm{cose}}{{\rm{c}}^2}\,{\rm{\theta }}]\,\,\cos \,\,{\rm{\theta }}\,d{\rm{\theta }}$ is equal to

${e^{\sin {\rm{\theta }}}}\,\,\left[ {\log \,\,\sin \,\,{\rm{\theta }}\,\, - \,\,{\rm{cosec}}\,\,{\rm{\theta }}} \right]\,\, + \,\,c$

$\smallint \,\,{e^{\sin {\rm{\theta }}}}\,\,[\log \,\,\sin \,\,{\rm{\theta }}\,\, + \,\,{\rm{cose}}{{\rm{c}}^2}\,\,{\rm{\theta }}]\,\, + \,\,c$

${e^{\sin {\rm{\theta }}}}\,\,\left[ {\log \,\,\sin \,\,{\rm{\theta }}\,\, + \,\,{\rm{cosec}}\,\,{\rm{\theta }}} \right]\,\, + \,\,c$

${e^{\sin \theta }}\,\,\left[ {\log \,\,\sin \,\,\theta \,\, - \,\,{\rm{cose}}{{\rm{c}}^2}\,\theta } \right]\,\, + \,\,c$

$\smallint \,\,x{e^{{x^2}}}\,\,dx$ is equal to

$\frac{{{e^x}}}{2}\,\, + \,\,c$

$ - \,\,\frac{{{e^x}}}{2}\,\, + \,\,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 811-840 of 995 questions

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811 Easy

811 of 995

Application +4 | -1

\int \frac{\sin x}{\sin (x-\alpha)} d x

=A x+B \log \sin (x-\alpha)+C

, then the value of

(A, B)

, is

View answer and explanation

Correct option: B

(\cos \alpha, \sin \alpha)

Option B (

\cos \alpha, \sin \alpha

) is correct because, after simplifying the integral using trigonometric identities, the coefficients of

x

and

\log \sin(x-\alpha)

are

\cos \alpha

and

\sin \alpha

respectively.

812 Easy

812 of 995

Analyzing +4 | -1

\int \frac{x^3-x^2+x-1}{x^5+1} d x

View answer and explanation

Correct option: C

\frac{1}{5} \ln \left(\frac{1+x^5}{(1+x)^5}\right)+c

Option C (

\frac{1}{5} \ln \left(\frac{1+x^5}{(1+x)^5}\right)+c

) is correct because the integral can be simplified and then integrated term by term.

813 Easy

813 of 995

Analyzing +4 | -1

\int\left(x+\sqrt{1+x^2}\right)^n d x.

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

Then

(a+b)

is equal to

View answer and explanation

Correct option: C

Option C is correct because after evaluating the integral and comparing coefficients, the value of (a+b) is found to be 4.

814 Hard

814 of 995

Application +4 | -1

The value of

\int \limits_1^2 {\left\{ {\,\,f\,\left( {\,g\left( x \right)} \right)} \right\}^{ - 1}}f'\,\,\left( {\,g\left( x \right)} \right)g'\,\,\left( x \right)\,\,dx

, where

g\,\,\left( 1 \right) = g\,\,\left( 2 \right)

, is equal to

View answer and explanation

Correct option: C

Option D (None of these) is correct because the integral evaluates to 0 due to the given condition g(1) = g(2).

815 Easy

815 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B is correct because the integral is solved by substituting

e^x = t

and using partial fraction decomposition.

816 Easy

816 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{-x^n}{x^{m+n}+1}+C

Option C (

\frac{-x^n}{x^{m+n}+1}+C

) is correct because integrating by parts and simplifying the resulting terms leads to this expression.

817 Medium

817 of 995

Application +4 | -1

\int\left[x+{\sqrt{x^2+1}}^n d x \,(n \neq \pm 1)\right.

View answer and explanation

Correct option: A

\frac{1}{2}\left[\frac{\left(x+\sqrt{x^2+1}\right)^{n+1}}{n+1}+\frac{\left(x+\sqrt{x^2+1}\right)^{n-1}}{n-1}\right]+c

Option A (

\frac{1}{2}\left[\frac{\left(x+\sqrt{x^2+1}\right)^{n+1}}{n+1}+\frac{\left(x+\sqrt{x^2+1}\right)^{n-1}}{n-1}\right]+c

) is correct because by substituting

t=x+\sqrt{x^2+1}

and simplifying, we can evaluate the integral.

818 Easy

818 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 after simplifying and integrating,

f(x)

is found to be

\tan x

819 Medium

819 of 995

Understanding +4 | -1

\int \,\,\left[ {f\left( x \right)\,\,\mathrm{g''}\,\,\left( x \right)\,\, - \,\,f''\,\,\left( x \right)\,\,\mathrm{g}\,\,\left( x \right)} \right]\,\,dx

is equal to

View answer and explanation

Correct option: C

f\left( x \right)\mathrm{g'}\,\,\left( x \right)\,\, - \,\,f'\,\,\left( x \right)\,\,\mathrm{g}\,\,\left( x \right)

Option C (

f(x)\mathrm{g'}\,(x)\, - \,f'\,(x)\,\mathrm{g}\,(x)

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

820 Medium

820 of 995

Application +4 | -1

Let

f

be a positive differentiable function defined on

(0, \infty)

and

\phi(x)=\lim \limits_{{n \rightarrow \infty}}\left(\frac{f\left(x+\frac{1}{n}\right)}{f(x)}\right)^n

Then

\int \log _e(\phi(x)) d x

is equal to

View answer and explanation

Correct option: C

\log _e(f(x))+c

Option C (

\log _e(f(x))+c

) is correct because by simplifying the limit and integrating, we can find the expression for the integral.

821 Easy

821 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C is correct because the integral is solved using substitution, where (2-x)/(2+x) = t³, simplifying the integral to a power rule.

822 Medium

822 of 995

Application +4 | -1

The anti-derivative of

f\left( x \right) = \frac{1}{{3\,\, + \,\,5\,\,\sin \,\,x\,\, + \,\,3\,\,\cos \,\,x}}

, whose graph passes through the point

\left( {\mathrm{0, 0}} \right)

View answer and explanation

Correct option: B

\frac{1}{5}\,\,\left( {\log \,\,\left| {1\,\, + \,\,\frac{5}{3}\,\,\tan \,\,\left( {x/2} \right)\,} \right|} \right)

Option B (

\frac{1}{5}\,\,\left( {\log \,\,\left| {1\,\, + \,\,\frac{5}{3}\,{{\tan }\,\,\left( {x/2} \right)\}} \right|} \right)

) is correct because it is the result of integrating the given expression using the tangent half-angle substitution and applying the initial condition.

823 Easy

823 of 995

Analyzing +4 | -1

\int \frac{4 x+1}{x^2+3 x+2} d x=a \log |x+1|+b \log |x+2|+C

, then

View answer and explanation

Correct option: B

a+b=4

Option B is correct because after performing partial fraction decomposition and integrating, we find that

a + b = 4

824 Easy

824 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral simplifies to this form after a u-substitution.

825 Easy

825 of 995

Analyzing +4 | -1

\int \,\,\frac{{x + 2}}{{2{x^2} + \,\,6x\,\, + 5}}\,\,dx = P\,\,\int \,\,\frac{{4x + 6}}{{2{x^2} + \,\,6x\,\, + 5}}\,\,dx\,\, + \,\,\frac{1}{2}\int \,\,\frac{{dx}}{{2{x^2} + \,\,6x\,\, + 5}}

, then the value of

P

View answer and explanation

Correct option: C

\frac{1}{4}

Option C (

\frac{1}{4}

) is correct because by manipulating the integral, we can find that P equals

\frac{1}{4}

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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

826 Easy

826 of 995

Understanding +4 | -1

If it is known that at the point

x=1

two antiderivative of

f(x)=e^x

differ by 2 . The difference of these antiderivatives at

x=100

View answer and explanation

Correct option: C

Option C (2) is correct because the difference between two antiderivatives of the same function is a constant, and this constant is given as 2 at x=1, so it remains 2 at x=100.

827 Easy

827 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A,

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

, is the correct answer because it results from partial fraction decomposition and integration of the given expression.

828 Easy

828 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

-\frac{1}{\cos x+\sin x}+c

Option A is correct because the integral simplifies to

-\frac{1}{\cos x + \sin x} + c

after recognizing the denominator as

(\sin x + \cos x)^2

and using substitution.

829 Medium

829 of 995

Analyzing +4 | -1

\int \frac{d x}{\left(x^6+1\right)^2}-\frac{5}{6} \int \frac{d x}{x^6+1}

is equal to

View answer and explanation

Correct option: A

\frac{x}{6\left(x^6+1\right)}

The correct answer is A (

\frac{x}{6(x^6+1)}

) because after applying integration by parts and simplifying the expression, the result matches option A.

830 Easy

830 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

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

831 Easy

831 of 995

Understanding +4 | -1

\int x^x\left(1+\log _e x\right) d x

is equal to

View answer and explanation

Correct option: C

x^x+C

Option C is correct because the integral can be simplified by recognizing that

x^x(1 + \log_e x)

is the derivative of

x^x

832 Easy

832 of 995

Application +4 | -1

\int \,{e^{\sin \mathrm{\theta }}}\,[\log \,\,\sin \,\,\mathrm{\theta } + \,\,\mathrm{cose}{\mathrm{c}^2}\,\mathrm{\theta }]\,\,\cos \,\,\mathrm{\theta }\,d\mathrm{\theta }

is equal to

View answer and explanation

Correct option: C

{e^{\sin \mathrm{\theta }}}\,\,\left[ {\log \,\,\sin \,\,\mathrm{\theta }\,\, - \,\,\mathrm{cosec}\,\,\mathrm{\theta }} \right]\,\, + \,\,c

The correct answer is e^{(sin θ)} [log sin θ - cosec θ] + c because integrating by parts and simplifying leads to this result.

833 Easy

833 of 995

Application +4 | -1

\int \,\,x{e^{{x^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because by substituting x² = t, the integral is simplified and evaluated to be e^(x²)/2 + c.

834 Easy

834 of 995

Analyzing +4 | -1

\int \frac{1}{a^2 \sin ^2 x+b^2 \cos ^2 x} d x=\frac{1}{12} \tan ^{-1}(3 \tan x)+C,

then the maximum value of

a \sin x+b \cos x

, is

View answer and explanation

Correct option: B

\sqrt{40}

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

ab = 12

and

\frac{a}{b} = 3

, leading to the maximum value of

a \sin x + b \cos x

being

\sqrt{40}

835 Easy

835 of 995

Application +4 | -1

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

equals

View answer and explanation

Correct option: A

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

Option A is correct because the integral simplifies to sec^(-1)(x^x) + C after recognizing the derivative of x^x.

836 Easy

836 of 995

Understanding +4 | -1

\int \mathrm{cosec}\;\,x\,dx = f\left( x \right) + \mathrm{constant},\mathrm{\;then}\,\;f\left( x \right)

is equal to

View answer and explanation

Correct option: B

\log \,\,|\tan \,\,\left( {x/2} \right)|

Option B (

\log \,|\tan \,\left( {x/2} \right)|

) is correct because the integral of cosec(x) is equal to log|tan(x/2)| + constant.

837 Easy

837 of 995

Analyzing +4 | -1

I=\int x^{27}\left(6 x^2+5 x+4\right)\left(x^2+x+1\right)^6 d x=f(x)+C

, then

f(x)

is equal to

View answer and explanation

Correct option: A

\frac{1}{7}\left(x^6+x^5+x^4\right)^7

Option D (

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

) is correct because by using the substitution

x^2=t

and recognizing the pattern for integration, the integral simplifies to the given expression.

838 Medium

838 of 995

Analyzing +4 | -1

The integral

\int\left(\left(\frac{x}{2}\right)^{x}+\left(\frac{2}{x}\right)^{x}\right) \log _{2} x 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 cannot be solved using a simple substitution method, and the given options do not represent the correct antiderivative.

839 Easy

839 of 995

Understanding +4 | -1

\int x^x(1+\ln |x|) d x

is equal to

View answer and explanation

Correct option: C

x^x+c

Option C is correct because the integral of x^x(1 + ln|x|) dx is x^x + c, which can be verified by differentiation.

840 Medium

840 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 (

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

) is correct because it is the result of the integration after dividing by

\cos^2 x

and using a suitable substitution.

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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