If $\smallint g\,\,\left( x \right)\,\,dx = g\left( x \right)$, then $\smallint g\,\,\left( x \right)\,\,\left\{ {f\left( x \right)\,\, + \,\,f'\left( x \right)} \right\}dx$ is equal to

$g\,\,\left( x \right)\,\,f\left( x \right)\,\, - \,\,g\,\,\left( x \right)\,\,f'\left( x \right)\,\, + \,\,C$

$g\,\,\left( x \right)\,\,{f^2}\left( x \right)\,\, + \,\,C$

If $f\left( x \right) = \frac{{x + 2}}{{2x + 3}}$. Then, $\smallint \,{\left( {\frac{{f\left( x \right)}}{{{x^2}}}} \right)^{1/2}}dx$ is equal to $\frac{1}{{\sqrt 2 }}\,\,{\rm{g}}\left( {\frac{{1 + \sqrt {2f\left( x \right)} }}{{1 - \sqrt {2f\left( x \right)} }}} \right)\,\, - \,\,\sqrt {\frac{2}{3}} \,\,h\left( {\frac{{\sqrt {3\,f\left( x \right)} \,\, + \,\,\sqrt 2 }}{{\sqrt {3\,f\left( x \right)} \,\, - \,\,\sqrt 2 }}} \right) + \,\,c$ where

${\rm{g}}\,\left( x \right) = {\tan ^{ - 1}}x,\;\,\,h\left( x \right) = \log \,\,\left| {\,x\,} \right|$

${\rm{g}}\,\left( x \right) = \log \,\,\left| {\,x\,} \right|,\;\,\,h\left( x \right) = {\tan ^{ - 1}}\,\,x$

${\rm{g}}\left( x \right) = h\left( x \right) = {\tan ^{ - 1}}x$

$\smallint \,\,\frac{{\log \,\,\sqrt x }}{{3x}}\,\,\;dx$is equal to

$\frac{2}{3}\,\,{\left( {\log \,\,\,\sqrt x } \right)^2} + c$

$\frac{2}{3}\,\,{\left( {\log \,\,x} \right)^2} + \,\,c$

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

The value of $\smallint {e^{\sec x}}\,\, \cdot \,\,{\sec ^3}\,\,x\,\,\left( {{{\sin }^2}\,\,x\,\, + \,\,\cos \,\,x\,\, + \,\,\sin \,\,x\,\, + \,\,\sin \,\,x\,\,\cos \,\,x} \right)dx$ is

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

$\smallint \,\,\frac{{dx}}{{\sin \,\,\left( {x - a} \right)\,\,\sin \left( {x - b} \right)}}$ is

$\frac{1}{{\sin \,\,\left( {a - b} \right)}}\,\,\log \,\,\left| {\,\frac{{\sin \left( {x - a} \right)}}{{{\rm{sin}}\left( {x - b} \right)}}\,} \right|\,\, + \,\,c$

$\frac{{ - 1}}{{\sin \,\,\left( {a - b} \right)}}\,\,\log \,\,\left| {\,\frac{{\sin \left( {x - a} \right)}}{{\sin \left( {x - b} \right)}}\,} \right|\,\, + \,\,c$

$\log \,\,\sin \,\,\left( {x - a} \right)\,\,\sin \,\,\left( {x - b} \right)\,\, + \,\,c$

$\log \,\,\left| {\,\frac{{\sin \left( {x - a} \right)}}{{\sin \left( {x - b} \right)}}\,} \right|\,\, + \,\,c$

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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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211 Easy

211 of 995

Analyzing +4 | -1

\int \,\,\frac{{x + \,\,\sqrt[3]{{{x^2}}} + \,\,\sqrt[6]{x}}}{{x\,(1 + \,\,\sqrt[3]{x}}}\,\,dx

View answer and explanation

Correct option: A

\frac{3}{2}\,\,{x^{2/3}} + \,\,6\,\,{\tan ^{ - 1}}\,\,{x^{1/6}} + \,\,C

Option A (

\frac{3}{2}\,{x^{2/3}} + \,6\,{\tan ^{ - 1}}\,{x^{1/6}} + \,C

) is correct because after substitution and simplification, the integral can be easily solved.

212 Easy

212 of 995

Application +4 | -1

\int \frac{1}{\log a}\left(a^x \cos a^x\right) d x=

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the result of two substitutions and applying the integral of cosine.

213 Easy

213 of 995

Application +4 | -1

The integral

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

is equal to (where

C

is a constant for integration)

View answer and explanation

Correct option: C

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

Option C (

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

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

214 Easy

214 of 995

Application +4 | -1

The value of

\int \frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x} d x

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because after simplifying the integrand using trigonometric identities, the integral can be easily evaluated.

215 Easy

215 of 995

Analyzing +4 | -1

\int \,\,\frac{{\cos \,\,4x\,\, + \,\,1}}{{\cos \,\,x\,\, - \,\,\tan \,\,x}}\,\,\;dx = A\,\,\cos \,\,4x\,\, + \,\,B

,then

View answer and explanation

Correct option: B

A = - \,\,\frac{1}{8}

The correct answer is B (

A = - \,\,\frac{1}{8}

) because after simplification and integration, the coefficient of

\cos 4x

is -1/8.

216 Medium

216 of 995

Application +4 | -1

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

, then

\int\{f(x)\}^2 d x

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because by differentiating the given integral equation and solving the differential equation, we find f(x) and then integrate \{f(x)\} ^2.

217 Easy

217 of 995

Application +4 | -1

\int g\,\,\left( x \right)\,\,dx = g\left( x \right)

, then

\int g\,\,\left( x \right)\,\,\left\{ {f\left( x \right)\,\, + \,\,f'\left( x \right)} \right\}dx

is equal to

View answer and explanation

Correct option: C

g\,\,\left( x \right)\,\,f\left( x \right)\,\, + \,\,C

Option C (

g\,\,( x )\,\,f( x )\,\, + \,\,C

) is correct because it is derived by using integration by parts, where the integral of g(x) is g(x) itself.

218 Medium

218 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

The correct answer is (1/2)log((1 + tan x)/(1 - tan x)) + c because the integral simplifies to a standard form after trigonometric manipulation and substitution.

219 Medium

219 of 995

Analyzing +4 | -1

\int \frac{{\sin \,\,\theta \,\, - \,\,\cos \,\,\theta }}{{\left( {\sin \,\,\theta \,\, + \,\,\cos \,\,\theta } \right)\,\,\sqrt {\sin \,\,\theta \,\,\cos \,\,\theta \,\, + \,\,{{\sin }^2}\,\,\theta \,\,{{\cos }^2}\,\,\theta } }}d\theta = \mathrm{cose}{\mathrm{c}^{ - 1}}\,\,\left( {f\left( \theta \right)} \right) + C

, then

View answer and explanation

Correct option: A

f\left( \theta \right) = \sin \,\,2\theta \,\, + \,\,1

Option A (

f(\theta ) = \sin \,2\theta \, + \,1

) is correct because after simplifying and integrating, the result matches the given form with this

f(\theta)

220 Hard

220 of 995

Analyzing +4 | -1

f\left( x \right) = \frac{{x + 2}}{{2x + 3}}

. Then,

\int \,{\left( {\frac{{f\left( x \right)}}{{{x^2}}}} \right)^{1/2}}dx

is equal to

\frac{1}{{\sqrt 2 }}\,\,\mathrm{g}\left( {\frac{{1 + \sqrt {2f\left( x \right)} }}{{1 - \sqrt {2f\left( x \right)} }}} \right)\,\, - \,\,\sqrt {\frac{2}{3}} \,\,h\left( {\frac{{\sqrt {3\,f\left( x \right)} \,\, + \,\,\sqrt 2 }}{{\sqrt {3\,f\left( x \right)} \,\, - \,\,\sqrt 2 }}} \right) + \,\,c

where

View answer and explanation

Correct option: D

\mathrm{g}\left( x \right) = \log \,\,\left| {\,x\,} \right|,\,\,\;h\left( x \right) = \log \,\,\left| {\,x\,} \right|

Option C (

\frac{{{5^{{5^{{5^x}}}}}}}{{{{\left( {\log \,5} \right)}^3}}}\\ + \\,C

) is correct because the integral is solved using repeated substitution and the formula for integrating exponential functions.

221 Easy

221 of 995

Application +4 | -1

For what value of

a

and

b

, the equation

\int \frac{d x}{1+\sin x}=\tan \left(\frac{x}{2}+a\right)+b \text { holds good.}

View answer and explanation

Correct option: C

a=-\frac{\pi}{4}, b

is any arbitrary constant

Option C (

a=-\frac{\pi}{4}, b

is any arbitrary constant) is correct because the integral of

\frac{dx}{1+\sin x}

simplifies to

\tan(\frac{x}{2} - \frac{\pi}{4}) + c

222 Easy

222 of 995

Application +4 | -1

\int e^{2 x^2+\ln x} d x

is equal to

View answer and explanation

Correct option: A

\frac{e^{2 x^2}}{4}+c

Option A (

\frac{e^{2 x^2}}{4}+c

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

223 Easy

223 of 995

Application +4 | -1

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

, then

f(x)=

View answer and explanation

Correct option: A

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

Option A,

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

, is the correct answer because the integral of

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

is indeed

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

224 Easy

224 of 995

Application +4 | -1

\int \,\,\frac{{\log \,\,\sqrt x }}{{3x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it results from direct integration after substitution.

225 Easy

225 of 995

Analyzing +4 | -1

The value of

\int {e^{\sec x}}\,\, \cdot \,\,{\sec ^3}\,\,x\,\,\left( {{{\sin }^2}\,\,x\,\, + \,\,\cos \,\,x\,\, + \,\,\sin \,\,x\,\, + \,\,\sin \,\,x\,\,\cos \,\,x} \right)dx

View answer and explanation

Correct option: C

{e^{\sec x}}\,\,\left( {\sec \,\,x\,\, + \,\,\tan \,\,x} \right)\,\, + \,\,C

Option C (

{e^{\sec x}}\,\,\left( {\sec \,\,x\,\, + \,\,\tan \,\,x} \right)\,\, + \,\,C

) is correct because it results from integrating the given expression using integration by parts and simplification.

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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

226 Easy

226 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

\frac{1}{b(a+b \cos x)}+c

Option B (

\frac{1}{b(a+b \cos x)}+c

) is correct because it is the result of a u-substitution and applying the power rule for integration.

227 Easy

227 of 995

Application +4 | -1

\int \,\,\frac{{{2^x}}}{{\sqrt {1\,\, - \,\,{4^x}} }}\,\,dx = K\,\,{\sin ^{ - 1}}\,\,\left( {{2^x}} \right)\,\, + \,\,C

, then

K

is equal to

View answer and explanation

Correct option: D

\frac{1}{{\log \,\,2}}

Option D (

\frac{1}{{\log \,2}}

) is correct because it results from applying a u-substitution and integrating the arcsine function.

228 Medium

228 of 995

Analyzing +4 | -1

\int \sin (101 x) \sin ^{99} x d x

equals

View answer and explanation

Correct option: A

\frac{1}{100} \sin (100 x)(\sin x)^{100}+C

Option A (

\frac{1}{100} \sin (100 x)(\sin x)^{100}+C

) is correct because by using the trigonometric identity

\sin(a+b) = \sin a \cos b + \cos a \sin b

and integration by parts, the integral simplifies to the given expression.

229 Easy

229 of 995

Analyzing +4 | -1

\int \frac{{dx}}{{{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){(1 + {x^6})^{\frac{1}{3}}} + C

where,

C

is a constant of integration, then the function

f(x)

is equal to

View answer and explanation

Correct option: B

- \frac{1}{{2{x^3}}}

Option B (

- \frac{1}{{2{x^3}}}

) is correct because after performing the integration and comparing with the given form,

f(x)

is found to be

- \frac{1}{{2{x^3}}}

230 Easy

230 of 995

Analyzing +4 | -1

\int \,\,\sqrt 2 \,\sqrt {1 + \sin \,\,x\,\;} dx = - \,4\,\,\cos \,\,\left( {ax + \,\,b} \right) + c

, then the value of

a,\,\,b

are respectively

View answer and explanation

Correct option: A

\frac{1}{2},\,\,\frac{\pi }{4}

Option A (

\frac{1}{2},\,\frac{\pi }{4}

) is correct because after simplifying the integral, the values of a and b can be identified by comparing the result with the given form.

231 Medium

231 of 995

Application +4 | -1

\int \,\,\frac{{dx}}{{\sin \,\,\left( {x - a} \right)\,\,\sin \left( {x - b} \right)}}

View answer and explanation

Correct option: A

\frac{1}{{\sin \,\,\left( {a - b} \right)}}\,\,\log \,\,\left| {\,\frac{{\sin \left( {x - a} \right)}}{{\mathrm{sin}\left( {x - b} \right)}}\,} \right|\,\, + \,\,c

Option A is correct because the integral is solved using trigonometric identities and logarithmic properties, resulting in the given expression.

232 Medium

232 of 995

Analyzing +4 | -1

For

\alpha, \beta, \gamma, \delta \in \mathbb{N}

, if

\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _e x

d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{8 x}+C

, where

e=\sum\limits_{{n=0}}^{\infty} \frac{1}{n !}

and

C

is constant of integration, then

\alpha+2 \beta+3 \gamma-4 \delta

is equal to

View answer and explanation

Correct option: B

Option B (4) is correct because after substituting

t = (x/e)^{2x}

, the integral simplifies and can be easily evaluated, leading to the values of

\alpha, \beta, \gamma, \delta

and thus the final result.

233 Medium

233 of 995

Application +4 | -1

\int \,\,\frac{1}{{\sqrt {{{\sin }^3}\,\,x\,\,\sin \,\,\left( {x + \alpha } \right)} }}\,\,dx

is equal to

View answer and explanation

Correct option: B

- \,2\,\,\mathrm{cosec}\,\,\alpha \,\,\sqrt {\cos \,\,\alpha + \sin \,\,\alpha \,\,\cot \,\,x} + C

Option B is correct because the integral simplifies to

-2 \,\,\mathrm{cosec}\,\,\alpha \,\,\sqrt {\cos \,\,\alpha + \sin \,\,\alpha \,\,\cot \,\,x} + C

after trigonometric manipulation and substitution.

234 Easy

234 of 995

Application +4 | -1

I=\int \frac{e^{\tan ^{-1}} x}{\left(1+x^2\right)}\left\{\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\} d x.

Then,

View answer and explanation

Correct option: B

I=e^{\tan ^{-1} x}\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+C

Option B is correct because the integral simplifies to

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

after a suitable substitution and simplification using trigonometric identities.

235 Easy

235 of 995

Application +4 | -1

\int \frac{1}{\log _x e} d x=

View answer and explanation

Correct option: C

x \log \left(\frac{x}{e}\right)+c

Option D (None of these) is incorrect. Option C (

x \log (\frac{x}{e})+c

) is correct because it is the result of integrating

\frac{1}{\log_x e}

using integration by parts and properties of logarithms.

236 Medium

236 of 995

Analyzing +4 | -1

\int \frac{\sin ^3 x}{\left(\cos ^4 x+3 \cos ^2 x+1\right) \tan ^{-1}(\sec x+\cos x)} d x

is equal to

View answer and explanation

Correct option: B

\log _e\left|\tan ^{-1}(\sec x+\cos x)\right|+C

Option B is correct because the integral simplifies to the logarithmic form of the inverse tangent function, resulting in

\log _e|\tan ^{-1}(\sec x+\cos x)|+C

237 Easy

237 of 995

Analyzing +4 | -1

I_n=\int \cot ^n x\, d x

, then

I_0+I_1+2

\left(I_2+I_3+\ldots+I_8\right)+I_9+I_{10}

equals to (where

\left.u=\cot x\right)

View answer and explanation

Correct option: B

-\left(u+\frac{u^2}{2}+\ldots+\frac{u^9}{9}\right)

Option B (-(

u+\frac{u^2}{2}+\ldots+\frac{u^9}{9}

)) is correct because using the reduction formula for the integral of cot^n(x) and summing the series leads to this expression.

238 Easy

238 of 995

Application +4 | -1

\int \frac{\sec ^2 x-7}{\sin ^7 x} d x

is equal to

View answer and explanation

Correct option: D

\tan x \operatorname{cosec}^7 x+c

Option D (

\tan x \operatorname{cosec}^7 x+c

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

239 Medium

239 of 995

Application +4 | -1

\int e^{\sin \theta}\left(\log \sin \theta+\operatorname{cosec}^2 \theta\right) \cos \theta d \theta

is equal to

View answer and explanation

Correct option: B

e^{\sin \theta}(\log \sin \theta-\operatorname{cosec} \theta)+C

Option B (

e^{\sin \theta}(\log \sin \theta-\operatorname{cosec} \theta)+C

) is correct because it results from applying integration by parts to the given integral.

240 Easy

240 of 995

Understanding +4 | -1

To find the value of

\int \frac{1+\log x}{x} d x

, the proper substitution is

View answer and explanation

Correct option: B

1+\log x=t

Option B (

1+\log x=t

) is correct because it simplifies the integral by making the derivative of the substitution equal to the remaining part of the integrand.

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