$\smallint \,{x^2}{\left( {ax\,\, + \,\,b} \right)^{ - 2}}dx$ is equal to

$\frac{2}{{{a^2}}}\,\,\left( {x\,\, - \,\,\frac{b}{a}\,\,\log \,\,\left( {ax\,\, + \,\,b} \right)} \right)\,\, + \,\,c$

$\smallint \,\,\frac{{{x^3}}}{{{{\left( {1 + {x^2}} \right)}^{1/3}}}}\,\,dx$ is equal to

$\frac{{20}}{3}\,\,{\left( {1 + {x^2}} \right)^{2/3}}\,\,\left( {2{x^2} - 3} \right) + C$

If $\smallint \frac{1}{{1\,\, + \,\,\sin \,\,x}}\,\,dx = \tan \,\,\left( {\frac{x}{2}\,\, + \,\,a} \right)\,\, + \,\,b$, then

$a = \frac{{5\,\pi }}{4},\;\,b \in R$

$\smallint \,\frac{{\left( {1\,\, + \,\,{x^4}} \right)}}{{{{\left( {1\,\, - \,\,{x^4}} \right)}^{3/2}}}}\,\,\;dx$ is equal to

$\frac{{2x}}{{\sqrt {1\,\, - \,\,{x^4}} }}\,\, + \,\,c$

$\frac{{ - 2x}}{{\sqrt {1\,\, - \,\,{x^4}} }}\,\, + \,\,c$

$\smallint \,\,\frac{{\sin \,\,x\;\,\,dx}}{{3 + 4\,\,{{\cos }^2}\,\,x}}$ is equal to

$\log \,(3\,\, + \,\,4\,\,{\cos ^2}\,\,x)\,\, + \,\,c$

$\frac{1}{{2\,\sqrt 3 }}\,\,{\tan ^{ - 1}}\left( {\frac{{\cos \,\,x}}{{\sqrt 3 }}} \right)\,\, + \,\,c$

The value of $\smallint \,\,\frac{{2\,\,dx}}{{\sqrt {1\,\, - \,\,4{x^2}} }}$ is

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

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

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

The value of the integral $\int \frac{1}{(x-3)^{6 / 7}(x+4)^{8 / 7}} d x$ is

$7\left(\frac{x-3}{x+4}\right)^{1 / 7}+C$

$7\left(\frac{x-3}{x+4}\right)^{6 / 7}+C$

$7\left(\frac{x+4}{x-3}\right)^{6 / 7}+C$

$\smallint \,\,\frac{1}{{x\;}}\,\,\left( {{{\log }_{ex}}\,\,e} \right)\,\,dx$ is equal to

${\log _e}\,\,({\log _e}\,\,ex\,\, - \,\,1)\,\, + \,\,c$

${\log _e}\,\,({\log _e}\,\,x\,\, - \,\,1)\,\, + \,\,c$

If $f(x)=\sqrt{x^2-a^2}$ then $\int \frac{x^2}{f(x)} \mathrm{d} x$ is

$x f(x)+\frac{a^2}{2} \log |x+f(x)|+C$

$\frac{x^2}{2} f(x)-\frac{a^2}{2} \log |x+f(x)|+C$

$\frac{x^2}{2} f(x)+a^2 \log |x-f(x)|+C$

The integral $\smallint \,\,\frac{{dx}}{{x\,\,{{\left( {1 + \,\,\log \,\,x} \right)}^2}}}$ is equal to

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

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

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

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

If ${I_n} = \smallint {\sin ^n}\,\,x\,dx$,then $n{I_n}\,\, - \,\,\left( {n - 1} \right){I_{n - 2}}$ equals

${\cos ^{n - 1}}\,\,x\,\,\sin x$

$ - \,\,{\cos ^{n - 1}}\,\,x\,\,\sin x$

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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 181-210 of 995 questions

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181 Easy

181 of 995

Understanding +4 | -1

\int \frac{f(x)}{[f(x)]^2} d x=

View answer and explanation

Correct option: A

-[f(x)]^{-1}+c

Option A (

-[f(x)]^{-1}+c

) is correct because the integral of

\frac{f'(x)}{[f(x)]^2}

is found by using substitution, where

f(x) = t

182 Easy

182 of 995

Analyzing +4 | -1

\int \,{x^2}{\left( {ax\,\, + \,\,b} \right)^{ - 2}}dx

is equal to

View answer and explanation

Correct option: B

\frac{2}{{{a^2}}}\,\,\left( {x\,\, - \,\,\frac{b}{a}\,\,\log \,\,\left( {ax\,\, + \,\,b} \right)} \right)\,\, - \,\,\frac{{{x^2}}}{{a\,\,\left( {ax\,\, + \,\,b} \right)}}\,\, + \,\,c

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

183 Hard

183 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\frac{3}{{20}}\,\,{\left( {1 + {x^2}} \right)^{2/3}}\,\,\left( {2{x^2} - 3} \right) + C

Option B (

\frac{3}{{20}}\left( {1 + {x^2}} \right)^{2/3}\left( {2{x^2} - 3} \right) + C

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

184 Easy

184 of 995

Application +4 | -1

\int \frac{\sqrt{\tan x}}{\sin x \cos x} d x=

View answer and explanation

Correct option: B

2 \sqrt{\tan x}+c

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

185 Easy

185 of 995

Application +4 | -1

The integral

\int {\frac{{dx}}{{{{\left( {x + 4} \right)}^{8/7}}{{\left( {x - 3} \right)}^{6/7}}}}}

is equal to (where

C

is a constant of integration)

View answer and explanation

Correct option: C

{\left( {\frac{{x - 3}}{{x + 4}}} \right)^{1/7}} + C

Option C is correct because the integral is solved by substitution, resulting in the given expression plus a constant of integration.

186 Easy

186 of 995

Application +4 | -1

\int \frac{1}{{1\,\, + \,\,\sin \,\,x}}\,\,dx = \tan \,\,\left( {\frac{x}{2}\,\, + \,\,a} \right)\,\, + \,\,b

, then

View answer and explanation

Correct option: A

a = - \frac{\pi }{4},\,\;b \in R

Option A (

a = - \frac{\pi }{4},\;b \in R

) is correct because it is obtained by integrating the given expression using trigonometric identities and comparing it with the target form.

187 Easy

187 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{x}{{\sqrt {1\,\, - \,\,{x^4}} }}\,\, + \,\,c

Option A (

\frac{x}{{\sqrt {1 - {x^4}} }} + c

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

188 Easy

188 of 995

Understanding +4 | -1

Assertion : The value of the integral

\int e^x\left[\tan x+\sec ^2 x\right] d x

\mathrm{e}^{\mathrm{x}} \tan \mathrm{x}+\mathrm{C}

Reason: The value of the integral

e^x\left\{f(x)+f^{\prime}(x)\right\} d x

\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})+\mathrm{C}

View answer and explanation

Correct option: A

Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.

Option A is correct because the assertion is true, and the reason correctly explains why the assertion is true. The integral of

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

is indeed

e^x f(x) + C

, and this applies to the given assertion with

f(x) = \tan x

189 Medium

189 of 995

Application +4 | -1

The value of

\int \nolimits_0^\pi |{\sin ^3}\,\,\theta |\,\,d\theta

View answer and explanation

Correct option: C

\frac{4}{3}

Option C is correct because the integral is evaluated by splitting the integral and using trigonometric substitution.

190 Easy

190 of 995

Application +4 | -1

\int\left(x^2+x\right)\left(x^{-8}+2 x^{-9}\right)^{1 / 10} d x

is equal to

View answer and explanation

Correct option: A

\frac{5}{11}\left(x^2+2 x\right)^{11 / 10}+C

Option A is correct because the integral simplifies to

\frac{5}{11}\left(x^2+2 x\right)^{11 / 10}+C

after a suitable substitution.

191 Easy

191 of 995

Application +4 | -1

\int \,\,\frac{{\sin \,\,x\;\,\,dx}}{{3 + 4\,\,{{\cos }^2}\,\,x}}

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is the correct result of the integration after substitution and applying the standard integral formula.

192 Medium

192 of 995

Application +4 | -1

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

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because substituting

x = \tan \theta

simplifies the integral, and integration by parts yields the correct result.

193 Medium

193 of 995

Analyzing +4 | -1

\int \frac{x^2+n(n-1)}{(x \sin x+n \cos x)^2} d x

is equal to

View answer and explanation

Correct option: D

\frac{-x^n \sec x}{x^n \sin x+n x^{n-1} \cos x}+\tan x+c

Option B (

\frac{{-\sec x}}{{x \sin x+n \cos x}}+\tan x+c

) is incorrect. Option D (

\frac{{-x^n \sec x}}{{x^n \sin x+n x^{n-1} \cos x}}+\tan x+c

) is the correct answer because after integration by parts and simplification, the expression matches this form.

194 Easy

194 of 995

Application +4 | -1

\int \frac{x}{x^2-4 x+8} \mathrm{~d} x=K \log \left(x^2-4 x+8\right)+

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

then the value of

K

View answer and explanation

Correct option: A

1 / 2

Option A (1/2) is correct because it is the coefficient of the logarithmic term in the antiderivative after completing the square and using substitution.

195 Easy

195 of 995

Application +4 | -1

The value of

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

View answer and explanation

Correct option: D

{\sin ^{ - 1}}\,\,\left( {2x} \right)\,\, + \,\,c

Option D (

{\sin ^{ - 1}}\,\,\left( {2x} \right)\, + \,c

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

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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

196 Medium

196 of 995

Analyzing +4 | -1

The value of the integral

\int \frac{1}{(x-3)^{6 / 7}(x+4)^{8 / 7}} d x

View answer and explanation

Correct option: A

\left(\frac{x-3}{x+4}\right)^{1 / 7}+C

Option A is correct because after rewriting the integral and using a simple substitution, the integral evaluates to

(\frac{x-3}{x+4})^{1 / 7}+C

197 Easy

197 of 995

Application +4 | -1

\int \,\,\frac{1}{{x\;}}\,\,\left( {{{\log }_{ex}}\,\,e} \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

{\log _e}\,\,(1\,\, + \,\,{\log _e}\,\,x)\,\, + \,\,c

Option D (

{\log _e}\,(1\, + \,{\log _e}\,x)\, + \,c

) is correct because after substituting

{\log _e}x = t

, the integral becomes a simple logarithmic form.

198 Easy

198 of 995

Application +4 | -1

f(x)=\sqrt{x^2-a^2}

then

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

View answer and explanation

Correct option: D

\frac{x}{2} f(x)+\frac{a^2}{2} \log |x+f(x)|+C

Option D (

\frac{x}{2} f(x)+\frac{a^2}{2} \log |x+f(x)|+C

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

199 Easy

199 of 995

Application +4 | -1

The integral

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

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

u = 1 + \log x

200 Easy

200 of 995

Analyzing +4 | -1

\int \frac{1}{\tan x+\cot x+\sec x+\operatorname{cosec} x} d x

is equal to

View answer and explanation

Correct option: D

None of these

The correct answer is D (None of these) because the integral evaluates to a different expression than the given options.

201 Medium

201 of 995

Application +4 | -1

f^{\prime}(x)=\tan ^{-1}(\sec x+\tan x), x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

and

f(0)=0

, then the value of

f(1)

View answer and explanation

Correct option: A

\frac{\pi+1}{4}

Option A is correct because the derivative simplifies to

f^{\prime}(x)=\frac{\pi}{4}+\frac{x}{2}

, and integrating with the given condition

f(0)=0

yields

f(1)=\frac{\pi+1}{4}

202 Easy

202 of 995

Application +4 | -1

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

equals

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 integrating the given expression after simplification and substitution.

203 Easy

203 of 995

Analyzing +4 | -1

Let

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

and

f(0)=0

. Then

f(1)

is equal to

View answer and explanation

Correct option: B

\log _e(1+\sqrt{2})-\frac{\pi}{4}

Option B is correct because after substituting

x = \tan \theta

and simplifying, the integral evaluates to

\log _e(1+\sqrt{2})-\frac{\pi}{4}

204 Easy

204 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because after dividing and using the arctangent integral, this is the result.

205 Easy

205 of 995

Application +4 | -1

The value of

\int \frac{1}{\sqrt{\sin ^3 x \cos ^5 x}} d x

, is

View answer and explanation

Correct option: A

\frac{-2}{\sqrt{\tan x}}+\frac{2}{3}(\tan x)^{3 / 2}+C

Option A (

\frac{{-2}}{{\sqrt{{\tan x}}}}+\frac{{2}}{{3}}(\tan x)^{{3 / 2}}+C

) is correct because the integral is solved by manipulating the integrand and using substitution.

206 Easy

206 of 995

Application +4 | -1

The value of

\sqrt{2} \int \frac{\sin x \,d x}{\sin \left(x-\frac{\pi}{4}\right)}

View answer and explanation

Correct option: B

x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+C

Option B (

x+\log \left|\sin \left(x-\frac{{\pi}}{{4}}\right)\right|+C

) is correct because it represents the result of the given integral after applying substitution and simplification.

207 Medium

207 of 995

Application +4 | -1

The value of

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

View answer and explanation

Correct option: A

\frac{-\log x}{x+1}+\log x-\log (x+1)

\frac{-\log x}{x+1}+\log x-\log (x+1)

Option A is correct because the integral of log(x)/(x+1)² dx is found using integration by parts, resulting in -log(x)/(x+1) + log(x) - log(x+1) + c.

208 Easy

208 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 the integral evaluates to this expression after a trigonometric substitution and simplification.

209 Easy

209 of 995

Application +4 | -1

{I_n} = \int {\sin ^n}\,\,x\,dx

,then

n{I_n}\,\, - \,\,\left( {n - 1} \right){I_{n - 2}}

equals

View answer and explanation

Correct option: C

- \,\,{\sin ^{n - 1}}\,\,x\,\,\cos x

Option C is correct because it is the result of applying integration by parts to the integral of sin^n(x).

210 Easy

210 of 995

Application +4 | -1

The value of

\int \frac{d x}{\sqrt{4 x-3-x^2}}

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the result of completing the square and using the standard integral formula for arcsine.

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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