$\smallint x\,\,\left\{ {\,f\,\left( {{x^2}} \right){\rm{g''}}\,\,\left( {{x^2}} \right)\,\, - \,\,f''\left( {{x^2}} \right)\,\,{\rm{g}}\,\,\left( {{x^2}} \right)} \right\}dx$ is equal to

$\frac{1}{2}\,\,\left\{ {\,f\left( {{x^2}} \right)\,\,{\rm{g'}}\,\,\left( {{x^2}} \right)\,\, - \,\,{\rm{g}}\,\,\left( {{x^2}} \right)\,\,f'\,\,\left( {{x^2}} \right)} \right\}\,\, + \,\,c$

$f\,\,\left( {{x^2}} \right)\,\,{\rm{g'}}\,\,\left( {{x^2}} \right)\,\, - \,\,{\rm{g}}\left( {{x^2}} \right)\,\,f'\,\,\left( {{x^2}} \right)\,\, + \,\,c$

$\frac{1}{2}\,\,\left\{ {\,\,f\,\,\left( {{x^2}} \right)\,\,{\rm{g}}\,\,\left( {{x^2}} \right)\,f'\left( {{x^2}} \right)} \right\}\,\, + \,\,c$

If $\int \tan ^7 x \mathrm{~d} x=f(x)+C$ then

$f(x)$ is a polynomial of degree 8 in $\tan x$

$f(x)$ is a polynomial of degree 5 in $\tan x$

$f(x)$ is a polynomial of degree 6 in $\tan x$

$\smallint \,\,\frac{{{\rm{cosec}}\,\,x}}{{{{\cos }^2}\,\,\left( {1\,\, + \,\,\log \,\,\tan \,\,\frac{x}{2}} \right)}}\,\,dx$

${\sin ^2}\,\,\left[ {1 + \,\,\log \,\,\tan \,\,\frac{x}{2}} \right]\,\, + \,\,c$

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

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

$\log \,\,\left( {{e^{2x}} + 1} \right) - \,\,{\tan ^{ - 1}}\,\,\left( {{e^x}} \right) + C$

$\frac{1}{2}\,\,\log \,\,\left( {{e^{2x}} + \,\,1} \right) - {\tan ^{ - 1}}\,\,\left( {{e^x}} \right) + C$

$\smallint \,\,{x^{51}}\,\,\left( {{{\tan }^{ - 1}}\,\,x\,\, + \,\,{{\cot }^{ - 1}}\,\,x} \right)\,\,dx$

$\frac{{\pi {x^{52}}}}{{104}}\,\, + \,\,\frac{\pi }{2}\,\, + \,\,c$

$\frac{{{x^{52}}}}{{52}}\,\, + \,\,\frac{\pi }{2}\,\, + \,\,c$

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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 331-360 of 995 questions

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331 Easy

331 of 995

Application +4 | -1

The value of

\int \frac{\sin \alpha}{\sqrt{1+\cos \alpha}} \mathrm{d} \alpha

View answer and explanation

Correct option: B

-2 \sqrt{2} \cos (\alpha / 2)+C

Option B (

-2 \sqrt{2} \cos (\alpha / 2)+C

) is correct because the integral of the given expression simplifies to this result after substitution.

332 Medium

332 of 995

Application +4 | -1

\int \tan ^4 x d x=A \tan ^3 x+B \tan x+f(x)

, then

View answer and explanation

Correct option: A

A=\frac{1}{3}, B=-1, f(x)=x+C

Option A is correct because

\int \tan^4 x dx = \frac{\tan^3 x}{3} - \tan x + x + C

, which matches the form

A \tan^3 x + B \tan x + f(x)

with

A=\frac{1}{3}, B=-1, f(x)=x+C

333 Easy

333 of 995

Application +4 | -1

\int a^{3 x+3} d x=

View answer and explanation

Correct option: B

\frac{a^{3 x+3}}{3 \log a}+c

Option B is correct because the integral of

a^{3x+3}

\frac{a^{3x+3}}{3 \log a} + c

after applying the substitution method.

334 Medium

334 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\,\,\left\{ {\,f\left( {{x^2}} \right)\,\,\mathrm{g'}\,\,\left( {{x^2}} \right)\,\, - \,\,\mathrm{g}\,\,\left( {{x^2}} \right)\,\,f'\,\,\left( {{x^2}} \right)} \right\}\,\, + \,\,c

Option C (

\frac{1}{2}\left{ f\left( {{x^2}} \right)\g'\left( {{x^2}} \right) - g\left( {{x^2}} \right)\f'\left( {{x^2}} \right)} \right} + c

) is correct because it is the result of applying integration by parts and substitution to the given integral.

335 Easy

335 of 995

Application +4 | -1

\int \frac{3 \sin x+2 \cos x}{3 \cos x+2 \sin x} d x=a x+b \log |3 \cos x+2 \sin x|+C

, then

View answer and explanation

Correct option: B

a=\frac{12}{13}, b=-\frac{5}{13}

Option B (

a=\frac{12}{13}, b=-\frac{5}{13}

) is correct because the values of a and b are obtained by comparing coefficients after expressing the numerator in terms of the derivative and the original denominator.

336 Easy

336 of 995

Application +4 | -1

\int \tan ^7 x \mathrm{~d} x=f(x)+C

then

View answer and explanation

Correct option: C

f(x)=\frac{1}{6} \tan ^6 x-\frac{1}{4} \tan ^4 x+\frac{1}{2} \tan ^2 x+\log |\cos x|+C

Option C is correct because the integral of tan⁷(x) can be found by using substitution and polynomial division.

337 Easy

337 of 995

Application +4 | -1

\int \,\,\frac{{\mathrm{cosec}\,\,x}}{{{{\cos }^2}\,\,\left( {1\,\, + \,\,\log \,\,\tan \,\,\frac{x}{2}} \right)}}\,\,dx

View answer and explanation

Correct option: B

\tan \,\,\left[ {1 + \,\,\log \,\,\tan \,\,\frac{x}{2}} \right]\,\, + \,\,c

Option B is correct because after the trigonometric simplification and substitution, the integral is a standard form, resulting in tan[1 + log tan(x/2)] + c.

338 Easy

338 of 995

Application +4 | -1

\int \frac{1-x^2}{\left(1+x^2\right) \sqrt{1+x^4}} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{\sqrt{2}} \sin ^{-1}\left\{\frac{\sqrt{2} x}{x^2+1}\right\}+C

Option B is correct. By dividing the numerator and denominator by

x^2

and using a substitution, the integral can be simplified to a standard inverse sine integral.

339 Easy

339 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

\ln \left(1-e^{-x}\right)+c

Option A is correct because the integral of

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

can be solved by rewriting the integrand as

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

and using substitution.

340 Medium

340 of 995

Application +4 | -1

\int \sin ^{-1} x d x

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the result of integrating

\sin^{-1}x

using integration by parts.

341 Medium

341 of 995

Application +4 | -1

\int \frac{1}{\cos ^2 x(1-\tan x)^2} d x=

View answer and explanation

Correct option: B

\frac{1}{1-\tan x}+c

Option B is correct because the integral simplifies to

\frac{{1}}{{1-\tan x}}+c

after substitution and integration.

342 Medium

342 of 995

Application +4 | -1

\int \frac{d x}{x\left(\ln ^2 x+4 \ln x-1\right)}

is equal to

View answer and explanation

Correct option: D

\frac{1}{2 \sqrt{5}} \ln \left|\frac{\ln x+2-\sqrt{5}}{\ln x+2+\sqrt{5}}\right|+c

Option D (

\frac{{1}}{{2 \sqrt{{5}}}} \ln \left| {\frac{{\ln x + 2 - \sqrt{{5}}}}{{\ln x + 2 + \sqrt{{5}}}}} \right| + c

) is correct because the integral can be solved by substitution and partial fraction decomposition.

343 Medium

343 of 995

Application +4 | -1

\int e^{\tan ^{-1} x}\left(1+\frac{x}{1+x^2}\right) 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 simplifies to this form after substitution and recognizing the pattern of

e^x(f(x) + f'(x))

344 Medium

344 of 995

Application +4 | -1

Let

\int e^x\left\{f(x)-f^{\prime}(x)\right\} d x=\phi(x)

. Then,

\int e^x f(x) d x

is equal to

View answer and explanation

Correct option: C

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

Option C is correct because it involves using the given integral and the known integral of e^x(f(x) + f'(x)) to find the required integral.

345 Easy

345 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of integrating the given expression after partial fraction decomposition.

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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

346 Easy

346 of 995

Application +4 | -1

\int \frac{x^{5 / 2}}{\sqrt{1+x^7}} d x

, is

View answer and explanation

Correct option: A

\frac{2}{7} \log \left|x^{7 / 2}+\sqrt{1+x^7}\right|+C

Option A is correct because the integral is solved using a substitution method, resulting in the given logarithmic expression.

347 Easy

347 of 995

Analyzing +4 | -1

\int \tan x \tan 2 x \tan 3 x \,d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{3} \log |\sec 3 x|-\frac{1}{2} \log |\sec 2 x|-\log |\sec x|+C

Option B is correct because using the identity

\tan 3x = \tan(2x+x)

and simplifying the integral, we get

\frac{1}{3} \log |\sec 3 x|-\frac{1}{2} \log |\sec 2 x|-\log |\sec x|+C

348 Easy

348 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the result of direct integration after separating the original integral into two simpler integrals.

349 Medium

349 of 995

Analyzing +4 | -1

\lim \limits_{{x \rightarrow 0}} \frac{f(x)}{x^2}

exists finitely and

\lim \limits_{{x \rightarrow 0}}\left(1+x+\frac{f(x)}{x}\right)^{1 / x}=e^3

, then

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

is equal to

View answer and explanation

Correct option: A

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

Option A is correct because after evaluating the limit and integrating, the result matches the given expression

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

350 Medium

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

351 Easy

351 of 995

Application +4 | -1

\int \log x d x=

View answer and explanation

Correct option: B

x \log x-x+c

Option B (

x \log x-x+c

) is correct because it is the result of integrating

\log x

using integration by parts.

352 Easy

352 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

(\cos \alpha-\sin \alpha)(x-\alpha)+(\cos \alpha+\sin \alpha) \log |\sin (x-\alpha)|+C

Option A is correct because after substituting

t = x - \alpha

and simplifying, the integral evaluates to the expression in option A.

353 Medium

353 of 995

Application +4 | -1

\int \,\,{x^{51}}\,\,\left( {{{\tan }^{ - 1}}\,\,x\,\, + \,\,{{\cot }^{ - 1}}\,\,x} \right)\,\,dx

View answer and explanation

Correct option: A

\frac{{{x^{52}}}}{{52}}\,\,\left( {{{\tan }^{ - 1}}\,\,x\,\, + \,\,{{\cot }^{ - 1}}\,\,x} \right)\,\, + \,\,c

Option A (

\frac{{{x^{52}}}}{{52}}\,\left( {{{\tan }^{ - 1}}\,x\, + \,{{\cot }^{ - 1}}\,x} \right)\, + \,c

) is correct because it correctly uses the identity

\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}

and integrates the resulting expression.

354 Easy

354 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the result of expanding the square and integrating each term.

355 Medium

355 of 995

Analyzing +4 | -1

\int \frac{\cos 4 x-1}{\cot x-\tan x} d x

is equal to

View answer and explanation

Correct option: D

\frac{1}{2} \ln \cos 2 x-\frac{\cos ^2 2 x}{4}+c

Option D is correct because after simplifying the integrand using trigonometric identities and substitution, the integral evaluates to the given expression.

356 Easy

356 of 995

Analyzing +4 | -1

\int \frac{x^2+4}{x^4+16} d x=\frac{1}{a} \tan ^{-1}\left(\frac{x^2-4}{a x}\right)+C

, then

a=

View answer and explanation

Correct option: B

2 \sqrt{2}

Option B (

2 \sqrt{2}

) is correct because after simplifying the integral, the value of 'a' is found to be

2 \sqrt{2}

357 Medium

357 of 995

Application +4 | -1

\int \frac{x^5 d x}{\sqrt{\left(1+x^3\right)}}=

View answer and explanation

Correct option: D

\frac{2}{9} \sqrt{\left(1+x^3\right)}\left(x^3-2\right)+c

Option D (

\frac{{2}}{{9}} \sqrt{{(1+x^3)}}(x^3-2)+c

) is correct because it results from correct substitution and integration of the given function.

358 Easy

358 of 995

Application +4 | -1

The value of

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

, is equal to

View answer and explanation

Correct option: A

\frac{1}{2} \sin ^{-1}\left(\frac{x}{a}\right)+\frac{1}{2} \log \left|x+\sqrt{a^2-x^2}\right|+C_1

Option A (

\frac{1}{2} \sin ^{-1}(\frac{x}{a})+\frac{1}{2} \log |x+\sqrt{a^2-x^2}|+C_1

) is correct because it is the result of the integral after applying the trigonometric substitution

x = a \sin \theta

and simplifying.

359 Easy

359 of 995

Application +4 | -1

\int \frac{1}{x^2+4 x+13} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{3} \tan ^{-1}\left(\frac{x+2}{3}\right)

Option B,

\frac{1}{3} \tan ^{-1}\left(\frac{x+2}{3}\right)

, is correct because completing the square and using the arctangent integral formula leads to this result.

360 Easy

360 of 995

Analyzing +4 | -1

\int \,\,\frac{{\left( {{x^3} + 3{x^2} + 3x + 1} \right)}}{{{{\left( {x + 1} \right)}^5}}}\,\,\;dx

is equal to

View answer and explanation

Correct option: A

- \,\,\frac{1}{{\left( {x + 1} \right)}}\,\, + \,\,c

Option A (

\ - \,\frac{1}{{(x + 1)}}\, + \,c

) is correct because the integrand simplifies to

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

, and its integral is

\ - \,\frac{1}{{(x + 1)}}\, + \,c

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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