$\int \frac{e^x d x}{\cos h x+\sin h x}$ is

$\frac{1}{2} \mathrm{e}^{2 \mathrm{x}}+\mathrm{c}$

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$

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

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

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

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

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

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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 151-180 of 746 questions

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151 Easy

151 of 746

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.

152 Easy

152 of 746

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.

153 Easy

153 of 746

Application +4 | -1

\int \frac{e^x d x}{\cos h x+\sin h x}

View answer and explanation

Correct option: B

x+c

Option B is correct because simplifying the denominator using the definitions of cosh x and sinh x leads to a simple exponential function, which integrates to x + c.

154 Easy

154 of 746

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).

155 Easy

155 of 746

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.

156 Easy

156 of 746

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.

157 Easy

157 of 746

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.

158 Easy

158 of 746

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.

159 Easy

159 of 746

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.

160 Easy

160 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\log (x+\log \sec x)+c

Option A is correct because the derivative of

x + \log(\sec x)

1 + \tan x

, making the integral a simple logarithmic form.

161 Easy

161 of 746

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.

162 Easy

162 of 746

Application +4 | -1

\int e^{\log_5 x} d x=

View answer and explanation

Correct option: B

\frac{x^{\log_5 5e}}{\log _5 5 e}

Option B is correct because the integral of

e^{\log_5 x}

simplifies to

\frac{x^{\log_5 5e}}{\log_5 5 e}

using properties of logarithms and integration.

163 Easy

163 of 746

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.

164 Easy

164 of 746

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

165 Easy

165 of 746

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.

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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

166 Easy

166 of 746

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

167 Easy

167 of 746

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.

168 Easy

168 of 746

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.

169 Easy

169 of 746

Application +4 | -1

\mathrm{I}=\int_{-3}^2(|\mathrm{x}+1|+|\mathrm{x}+2|+|\mathrm{x}-1|) \mathrm{dx}

, then

\mathrm{I}

equals:

View answer and explanation

Correct option: A

\frac{31}{2}

Option A is correct because the integral is solved by using a trigonometric substitution, which simplifies the expression and allows for direct integration.

170 Easy

170 of 746

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.

171 Easy

171 of 746

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.

172 Easy

172 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

x^3 \tan \frac{1}{x}+c

Option C (

x^3 \tan \frac{{1}}{{x}}+c

) is correct because the integral is solved using integration by parts and simplification.

173 Easy

173 of 746

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}}}

174 Easy

174 of 746

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.

175 Easy

175 of 746

Application +4 | -1

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

, then

f(x)

can be

View answer and explanation

Correct option: D

\sin x

Option D (

\sin x

) is correct because when

f(x) = \sin x

, the integral

\int f(x) \cos x dx

results in

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

176 Easy

176 of 746

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.

177 Easy

177 of 746

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.

178 Easy

178 of 746

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.

179 Easy

179 of 746

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.

180 Easy

180 of 746

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.

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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