$\smallint \,\,{e^{3\,\,\log x}}{\left( {{x^4} + 1} \right)^{ - 1}}\,\,dx$ is equal to

${e^{3\,\,\log x}}\,\, + \,\,c$

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

The value of $\int \frac{\cos x}{\sin x+\cos x} \mathrm{~d} x$ is

$\frac{x^2}{2}+\log |\sin x+\cos x|+C$

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

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

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

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

$\frac{{{a^{\sqrt x }}}}{{\log \,\,a}}\,\, + \,\,C$

$2{a^{\sqrt x }}\,\, \cdot \,\,\log \,\,a\,\, + \,\,C$

$\smallint \,\,\frac{{{3^x}}}{{\sqrt {{9^x}\,\, - \,\,1} }}\,\,\;dx$ is equal to

$\frac{1}{{\log \,\,3}}\,\,\log \,\,\left| {\,{9^x}\,\, + \,\,\sqrt {{9^x} - 1\,} } \right|\,\, + \,\,c$

$\frac{1}{{\log \,\,9}}\,\,\log \,\,\left| {\,{3^x}\,\, + \,\,\sqrt {{9^x} - 1\,} } \right|\,\, + \,\,c$

$\frac{1}{{\log \,\,9}}\,\,\log \,\,\left| {\,{3^x}\,\, - \,\,\sqrt {{9^x} - 1\,} } \right|\,\, + \,\,c$

$\smallint \,\,{e^x}\,\,\frac{{\left( {x - 1} \right)}}{{{x^2}}}\;\,\,dx$ is equal to

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Easy

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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 746 questions

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331 Easy

331 of 746

Application +4 | -1

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

is equal to

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 it is the result of simplifying the integral and applying a u-substitution.

332 Easy

332 of 746

Application +4 | -1

The value of

\int \frac{\cos x}{\sin x+\cos x} \mathrm{~d} x

View answer and explanation

Correct option: B

\frac{1}{2}(x+\log |\sin x+\cos x|)+C

Option B is correct because the integral can be solved by splitting the numerator and using logarithmic integration.

333 Easy

333 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

{\tan ^{ - 1}}\,\,\left( {\sqrt x } \right) + c

Option B (

{\tan ^{ - 1}}\,\,\left( {\sqrt x } \right) + c

) is correct because it is obtained by using a simple u-substitution with

u = \sqrt{x}

334 Easy

334 of 746

Analyzing +4 | -1

\int \frac{1}{x+x^5} d x=f(x)+C

, then the value of

\int \frac{x^4}{x+x^5} d x

View answer and explanation

Correct option: A

\log x-f(x)+C

The correct answer is A (

\log x-f(x)+C

) because by manipulating the integrand and using the given information, the integral can be expressed in terms of

f(x)

and

\log x

335 Easy

335 of 746

Application +4 | -1

\int \frac{\left\{x+\sqrt{x^2+1}\right\}^n}{\sqrt{x^2+1}} d x

is equal to

View answer and explanation

Correct option: B

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

Option B is correct. The integral is solved by recognizing that the derivative of

x + \sqrt{x^2 + 1}

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

, allowing for a direct substitution and integration.

336 Easy

336 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

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

337 Easy

337 of 746

Application +4 | -1

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

equals

View answer and explanation

Correct option: A

\frac{1}{2}\,\,\log \,\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{{12}}} \right) + c

Option A is correct because the integral of the given function simplifies to a logarithmic tangent function with the correct phase shift.

338 Easy

338 of 746

Application +4 | -1

\int \,\,\frac{{{a^{\sqrt x }}}}{{\sqrt x }}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{2{a^{\sqrt x }}}}{{\log \,\,a}}\,\, + \,\,C

The correct answer is B (

\frac{{2{a^{\sqrt x }}}}{{\log \,\,a}}\, + \,\,C

) because after applying the substitution method and integrating, the result matches option B.

339 Easy

339 of 746

Application +4 | -1

\mathrm{I}=\int \frac{\mathrm{x}^5}{\sqrt{1+\mathrm{x}^3}} \mathrm{dx}

, then

\mathrm{I}

is equal to:

View answer and explanation

Correct option: B

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

Option B is correct because the integral is solved by substitution, letting

1+x^3 = t^2

and simplifying the resulting expression.

340 Easy

340 of 746

Application +4 | -1

\int \sqrt[3]{x}\left(1+\sqrt[3]{x^4}\right)^{1 / 7} d x=\alpha\left(1+\sqrt[3]{x^4}\right)^\beta+c

, then

\alpha+\beta^{-1}

View answer and explanation

Correct option: D

\frac{49}{32}

Option D (

\frac{49}{32}

) is correct because after evaluating the integral and comparing coefficients,

\alpha+\beta^{-1}

equals this value.

341 Easy

341 of 746

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the result of the integration after a suitable substitution.

342 Easy

342 of 746

Application +4 | -1

\int \frac{a^{\sqrt{x}}}{\sqrt{x}} d x=

View answer and explanation

Correct option: B

2 a^{\sqrt{x}} \log _a e+c

Option B (

2 a^{\sqrt{{x}}} \log _a e+c

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

\int a^x dx = \frac{{a^x}}{{\log_e a}} + C = a^x \log_a e + C

343 Easy

343 of 746

Application +4 | -1

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

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because integrating by parts simplifies the expression, leading to the given result.

344 Easy

344 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D is correct because the integral is solved using substitution, resulting in

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

345 Easy

345 of 746

Application +4 | -1

I=\int \frac{\cos 2 x-\cos 2 \alpha}{\sin x-\sin \alpha} d x

, then

I

equals

View answer and explanation

Correct option: B

2 \cos x-2 x \sin \alpha+C

Option B (

2 \cos x-2 x \sin \alpha+C

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

346 Easy

346 of 746

Application +4 | -1

\int \nolimits_0^1 \,\,\frac{{{x^7}}}{{\sqrt {1\,\, - \,\,{x^4}} }}\;\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{1}{3}

Option B (

\frac{1}{3}

) is correct because it is the result of evaluating the definite integral using substitution and the Gamma function.

347 Easy

347 of 746

Understanding +4 | -1

Let

f(x)=\frac{1}{x} \ln \left(\frac{x}{e^x}\right)

then its primitive w.r.t.

x

View answer and explanation

Correct option: C

\frac{1}{2} \ln ^2 \mathrm{x}-\mathrm{x}+\mathrm{C}

Option C (

\frac{1}{2} \ln ^2 \mathrm{x}-\mathrm{x}+\mathrm{C}

) is correct because it is the result of integrating the given function after simplification using logarithm properties.

348 Easy

348 of 746

Analyzing +4 | -1

The value of

I=\int \frac{d x}{\left(a+b x^2\right) \sqrt{b-a x^2}}

, is

View answer and explanation

Correct option: A

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

Option A is correct because the integral evaluates to

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

after using a trigonometric substitution and simplification.

349 Easy

349 of 746

Application +4 | -1

The value of

\int \frac{\ell n |x|}{x \sqrt{1+\ell n |x|}} d x

equals:

View answer and explanation

Correct option: A

\frac{2}{3} \sqrt{1+\ell n|x|}(\ell n|x|-2)+C

Option A (

\frac{2}{3} \sqrt{1+\ell n|x|}(\ell n|x|-2)+C

) is correct because it results from a u-substitution and subsequent integration.

350 Easy

350 of 746

Application +4 | -1

\int \sec x \log (\sec x+\tan x) d x=

View answer and explanation

Correct option: B

\frac{1}{2}[\log (\sec x+\tan x)]^2+c

Option B is correct because the integral evaluates to

\frac{1}{2}[\log (\sec x+\tan x)]^2+c

after a simple substitution.

351 Easy

351 of 746

Analyzing +4 | -1

\int \frac{\cos 2 x \sin 4 x}{\cos ^4 x\left(1+\cos ^2 2 x\right)} d x=2 \log _e(1+\cos 2 x)-\log _e\left(1+\cos ^2 2 x\right)+f(x)+\mathrm{c}

then

f(x)

View answer and explanation

Correct option: A

\sec ^2 x

Option A is correct because the integral can be solved by trigonometric manipulation and partial fractions, leading to the given expression for f(x).

352 Easy

352 of 746

Application +4 | -1

\int 32 x^3(\log x)^2 d x

is equal to

View answer and explanation

Correct option: A

x^4\left\{8(\log x)^2-4(\log x)+1\right\}+c

Option A (

x^4\left\{8(\log x)^2-4(\log x)+1\right\}+c

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

353 Easy

353 of 746

Application +4 | -1

\int \frac{(x+1)(x+\log x)^2}{x} d x=

View answer and explanation

Correct option: C

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

Option C is correct because the integral simplifies to

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

after applying the substitution method.

354 Easy

354 of 746

Application +4 | -1

\int \,\,\frac{{{3^x}}}{{\sqrt {{9^x}\,\, - \,\,1} }}\,\,\;dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{{\log \,\,3}}\,\,\log \,\,\left| {\,{3^x}\,\, + \,\,\sqrt {{9^x} - 1\,} } \right|\,\, + \,\,c

Option A is correct because the integral can be solved by substituting

{3^x} = t

and using the standard integral formula.

355 Easy

355 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

\frac{1}{\log a} \sin ^{-1} a^x+c

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

356 Easy

356 of 746

Application +4 | -1

\int \,\,{e^x}\,\,\frac{{\left( {x - 1} \right)}}{{{x^2}}}\;\,\,dx

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the integral can be simplified and solved using the property

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

357 Easy

357 of 746

Application +4 | -1

\int \frac{1}{\sqrt{x}} \tan ^4 \sqrt{x} \sec ^2 \sqrt{x} d x=

View answer and explanation

Correct option: C

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

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

358 Easy

358 of 746

Application +4 | -1

The value of the integral

\int \limits_2^4 \,\,\frac{{\sqrt {{x^2} - \,\,4} }}{{{x^4}}}\,\,dx

, is

View answer and explanation

Correct option: B

\frac{{\sqrt 3 }}{{32}}

Option B (

\frac{{\sqrt 3 }}{{32}}

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

359 Easy

359 of 746

Application +4 | -1

\int e^x\left(1+\tan x+\tan ^2 x\right) d x=

View answer and explanation

Correct option: C

e^x \tan x+c

Option C is correct because the integral simplifies to

e^x \tan x + c

by recognizing the form

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

360 Easy

360 of 746

Application +4 | -1

\int {\frac{{\tan x}}{{\sqrt {\cos x} }}dx}

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

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

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