The anti-derivative of $\frac{{{2^x}}}{{\sqrt {1 - {4^x}} }}$ w.r.t.$x$ is

${\cos ^{ - 1}}\left( {{2^x}} \right){\log _2}e + k$

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

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

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

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

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

If $I = \smallint \,\,\frac{{{x^5}}}{{\sqrt {1\,\, + \,\,{x^3}} }}\,\,dx$, then $I$ is equal to

$\frac{2}{9}\,\,{\left( {1 + {x^3}} \right)^{\frac{5}{2}}}\,\, + \,\,\frac{2}{3}\,\,{\left( {1 + {x^3}} \right)^{\frac{3}{2}}}\,\, + \,\,c$

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

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

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

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

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

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

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

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

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

If $\int {{e^{\sec x}}} (\sec x\tan x{\rm{ }}f(x) + (sec{\rm{ }}x\tan x + {\sec ^2}x)){\rm{ }}dx = {e^{\sec x}}f(x) + C$, then a possible choice of $f(x)$ is

$x\sec x + \tan x + \frac{1}{2}$

$x\sec x + x\tan x - \frac{1}{2}$

$\sec x - x\tan x - \frac{1}{2}$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

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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 121-150 of 411 questions

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121 Medium

121 of 411

Analyzing +4 | -1

\int \sqrt{(x-3)}\left[\sin ^{-1}(\ln x)+\cos ^{-1}(\ln x)\right] d x

equals

View answer and explanation

Correct option: C

does not exist

Option C is correct because the domains of the two functions inside the integral do not overlap, so the integral does not exist.

122 Medium

122 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because by substituting

t = e^x + x^e

, the integral simplifies to

\frac{1}{e} \int \frac{dt}{t}

, which integrates to

\frac{1}{e} \ln(e^x + x^e) + c

123 Medium

123 of 411

Application +4 | -1

Let

x^2 \neq n \pi-1, n \in N

. Then, the value of

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

is equal to

View answer and explanation

Correct option: B

\log \left|\sec \left(\frac{x^2+1}{2}\right)\right|+C

Option B is correct because by using the trigonometric identity

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

, the integral simplifies to

\log \left|\sec \left(\frac{x^2+1}{2}\right)\right|+C

124 Medium

124 of 411

Application +4 | -1

The anti-derivative of

\frac{{{2^x}}}{{\sqrt {1 - {4^x}} }}

w.r.t.

x

View answer and explanation

Correct option: A

{\log _2}e \cdot {\sin ^{ - 1}}\left( {{2^x}} \right) + k

Option C (

{\cos ^2}x\left( {\frac{1}{2} - \log \cos x} \right) + k

) is correct because after integrating by parts and simplifying, the result matches this option.

125 Medium

125 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

- \,{e^x}\,\mathrm{cot}\,\,\left( {\frac{x}{2}} \right) + c

Option B (

\ - \,{e^x}\,\mathrm{cot}\,\,\left( {\frac{x}{2}} \right) + c

) is correct because after simplifying the integrand and applying integration by parts, the result matches this option.

126 Medium

126 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (-

(1+\frac{1}{x^4})^{1 / 4}+c

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

127 Medium

127 of 411

Application +4 | -1

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

, then

I

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

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

128 Medium

128 of 411

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

129 Medium

129 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\operatorname{ln} \left|\tan \left(\frac{\sin ^{-1} e^x}{2}\right)\right|+c

Option C is correct because the integral simplifies to

\ln \left|\tan \left(\frac{\sin ^{-1} e^x}{2}\right)\right|+c

after substitution and simplification using trigonometric identities.

130 Medium

130 of 411

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.

131 Medium

131 of 411

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.

132 Medium

132 of 411

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.

133 Medium

133 of 411

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.

134 Medium

134 of 411

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))

135 Medium

135 of 411

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.

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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 Medium

136 of 411

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

137 Medium

137 of 411

Application +4 | -1

I=\int \frac{\left(x+x^{\frac{2}{3}}+x^{\frac{1}{6}}\right)}{x\left(1+x^{\frac{1}{3}}\right)} d x

is equal to

View answer and explanation

Correct option: A

\frac{3}{2} x^{\frac{2}{3}}+6 \tan ^{-1}\left(x^{\frac{1}{6}}\right)+c

Option A is correct because the integral simplifies to

\frac{3}{2} x^{\frac{2}{3}}+6 \tan ^{-1}\left(x^{\frac{1}{6}}\right)+c

after substitution and simplification.

138 Medium

138 of 411

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.

139 Medium

139 of 411

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.

140 Medium

140 of 411

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.

141 Medium

141 of 411

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.

142 Medium

142 of 411

Analyzing +4 | -1

\int {{e^{\sec x}}} (\sec x\tan x\mathrm{ }f(x) + (sec\mathrm{ }x\tan x + {\sec ^2}x))\mathrm{ }dx = {e^{\sec x}}f(x) + C

, then a possible choice of

f(x)

View answer and explanation

Correct option: B

\sec x + \tan x + \frac{1}{2}

Option B (

\sec x + \tan x + \frac{1}{2}

) is correct because differentiating

e^{\sec x}f(x)

and comparing with the integrand allows us to identify

f(x)

143 Medium

143 of 411

Application +4 | -1

\int \frac{e^x(1+\sin x) d x}{1+\cos x}=e^x f(x)+c

, then

f(x)=

View answer and explanation

Correct option: C

\tan \frac{x}{2}

Option C (

\tan \frac{x}{2}

) is correct because when the integral is simplified, the function f(x) is identified as

\tan \frac{x}{2}

based on the form

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

144 Medium

144 of 411

Analyzing +4 | -1

\int {\frac{{3\tan \left( {x - \frac{\pi }{4}} \right)}}{{{{\cos }^2}x\sqrt {{{\tan }^3}x + {{\tan }^2}x + \tan x} }}dx = k\,\,{{\tan }^{ - 1}}\left( {\sqrt {\tan x + 1 + \cot x} } \right) + C,}

then the value of

k

is: [where

C

is constant of integration].

View answer and explanation

Correct option: C

Option C, 6, is correct because after substitution and integration, the value of k is found to be 6.

145 Medium

145 of 411

Application +4 | -1

\int \frac{d x}{\sqrt{1-\tan ^2 x}}=\frac{1}{\lambda} \sin ^{-1}(\lambda \sin x)+C

, then

\lambda=

View answer and explanation

Correct option: A

\sqrt{2}

Option A (

\sqrt{2}

) is correct because by manipulating the integrand and using a trigonometric substitution, the integral can be solved, revealing that

\lambda = \sqrt{2}

146 Medium

146 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{1}{2} \ln ^2\left(x+\sqrt{x^2+1}\right)+c

Option C (

\frac{1}{2} \ln ^2(x+\sqrt{x^2+1})+c

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

147 Medium

147 of 411

Analyzing +4 | -1

The integral

\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x

is equal to

View answer and explanation

Correct option: A

\frac{1}{2} \log _e\left|\frac{\tan \left(\frac{x}{2}+\frac{\pi}{12}\right)}{\tan \left(\frac{x}{2}+\frac{\pi}{6}\right)}\right|+C

Option A (

\frac{{1}}{{2}} \log _e\left| {\frac{{\tan \left( {\frac{{x}}{{2}} + \frac{{\pi }}{{12}}} \right)}}{{\tan \left( {\frac{{x}}{{2}} + \frac{{\pi }}{{6}}} \right)}}} \right| + C

) is correct because after simplification and integration, the given integral evaluates to this expression.

148 Medium

148 of 411

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B (

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

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

149 Medium

149 of 411

Application +4 | -1

\int \frac{f(x) g^{\prime}(x)+f^{\prime}(x) g(x)}{f(x) \,g(x)}\{\log f(x)+\log g(x)\} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}[\log \{f(x)\, g(x)\}]^2+C

Option B is correct because the integral simplifies to a form that can be solved by recognizing the derivative of log(f(x)g(x)).

150 Medium

150 of 411

Analyzing +4 | -1

\int \,\,uv''\,dx = uv' - \,\,vu' + a

, then

a

is equal to

View answer and explanation

Correct option: A

\int u''v\;\,\,dx

Option A is correct because after applying integration by parts twice,

a

is equal to

\int u''v\;\,\,dx

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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