$\smallint 32\,\,{x^3}{(\log \,\,x)^2}\,\,\;dx$ is equal to

${x^4}\{ \,8\,{(\log \,\,x)^2}\,\, - \,\,4\log \,x\,\} + \,c$

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

$\smallint \,\,{\rm{cose}}{{\rm{c}}^4}\,\,x\,\,\;dx$ is equal to

$\tan \,\,x\,\, + \,\,\frac{{{{\tan }^3}\,\,x}}{3}\,\, + \,\,c$

$ - \tan \,\,x\,\, - \,\,\frac{{{{\tan }^3}\,\,x}}{3}\,\, + \,\,c$

If $d\,\,\left[ {\,f\,\,\left( x \right)} \right] = {e^{\tan x}}\,\,{\sec ^2}\,\,x\,\,\;dx,\;\,\,{\rm{then}}\;\,\,f\left( x \right)$ is equal to

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

The value of $\smallint \,\,\frac{{{x^2}}}{{1\,\, + \,\,{x^6}}}\,\,dx$ is

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

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

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

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

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

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

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

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

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

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

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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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121 Easy

121 of 746

Application +4 | -1

\int 32\,\,{x^3}{(\log \,\,x)^2}\,\,\;dx

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because the integral is solved using integration by parts twice.

122 Easy

122 of 746

Application +4 | -1

\int \,\,\mathrm{cose}{\mathrm{c}^4}\,\,x\,\,\;dx

is equal to

View answer and explanation

Correct option: C

- \,\,\cot \,\,x\,\, - \,\,\frac{{{{\cot }^3}\,\,x}}{3}\,\, + \,\,c

Option C (

- \,\cot \,x\, - \,\frac{{{{\cot }^3}\,x}}{3}\, + \,c

) is correct because it is the result of integrating

cosec^4(x)

by rewriting it as

cosec^2(x)(1 + cot^2(x))

123 Easy

123 of 746

Application +4 | -1

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

equals
Where

c

is an arbitrary constant

View answer and explanation

Correct option: B

\log \,\,\left| {\,\frac{x}{{x\,\, + \,\,1}}\,} \right| + c

Option B,

\log \,\,\left| {\,\frac{x}{{x\,\,+\,\,1}}\} \right| + c

, is correct because the integral of the given expression can be solved using partial fractions, resulting in this logarithmic form.

124 Easy

124 of 746

Application +4 | -1

f^{\prime}(x)=3 x^2 \sin \frac{1}{x}-x \cos \frac{1}{x}, x \neq 0, f(0)=0

then the value of

{f}\left(\frac{1}{\pi}\right)

View answer and explanation

Correct option: A

Option A (0) is correct because after integrating f'(x) and applying the initial condition f(0) = 0, we find that f(1/π) = 0.

125 Easy

125 of 746

Understanding +4 | -1

d\,\,\left[ {\,f\,\,\left( x \right)} \right] = {e^{\tan x}}\,\,{\sec ^2}\,\,x\,\,\;dx,\;\,\,\mathrm{then}\;\,\,f\left( x \right)

is equal to

View answer and explanation

Correct option: A

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

Option A (e^{(tan x)} + c) is correct because it is the antiderivative of e^{(tan x)} * sec²(x), found by recognizing that sec²(x) is the derivative of tan(x).

126 Easy

126 of 746

Application +4 | -1

\int x^2 e^{x^3} \cos \left(e^{x^3}\right) d x

is equal to

View answer and explanation

Correct option: C

\frac{1}{3} \sin \left(e^{x^3}\right)+C

Option C (

\frac{1}{3} \sin \left(e^{x^3}\right)+C

) is correct because after simplifying the integral using substitution, the result matches this option.

127 Easy

127 of 746

Application +4 | -1

The value of

\int \,\,\frac{{{x^2}}}{{1\,\, + \,\,{x^6}}}\,\,dx

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it results from a u-substitution where

u = x^3

, leading to the integral of

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

128 Easy

128 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A is correct because the integral can be solved by rewriting the integrand and using substitution, leading to the expression

\log(1+e^x) - x - e^{-x} + c

129 Easy

129 of 746

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

130 Easy

130 of 746

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.

131 Easy

131 of 746

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.

132 Easy

132 of 746

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.

133 Easy

133 of 746

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.

134 Easy

134 of 746

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.

135 Easy

135 of 746

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

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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

136 Easy

136 of 746

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.

137 Easy

137 of 746

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.

138 Easy

138 of 746

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C (

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

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

139 Easy

139 of 746

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.

140 Easy

140 of 746

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.

141 Easy

141 of 746

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.

142 Easy

142 of 746

Application +4 | -1

\int \tan ^3 2 x \sec 2 x d x=

View answer and explanation

Correct option: B

\frac{1}{6}\left[\sec ^3 2 x-3 \sec 2 x\right]

Option B (

\frac{1}{6}[\sec ^3 2 x-3 \sec 2 x]

) is correct because it is the result of substituting

\sec 2x = t

and integrating the resulting polynomial.

143 Easy

143 of 746

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.

144 Easy

144 of 746

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

145 Easy

145 of 746

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.

146 Easy

146 of 746

Application +4 | -1

I=\int \frac{\left(10 x^9+10^x \log _e 10\right)}{\left(x^{10}+10^x\right)} d x

is equal to

View answer and explanation

Correct option: D

\ln \left(10^x+x^{10}\right)+c

Option D (

\ln \left(10^x+x^{10}\right)+c

) is correct because the numerator is the derivative of the denominator, making it a simple u-substitution problem.

147 Easy

147 of 746

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.

148 Easy

148 of 746

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}

149 Easy

149 of 746

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.

150 Easy

150 of 746

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.

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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