If ${I_n} = \smallint {\tan ^n}\,\,x\,dx$ for $n \ge 2$, then ${I_n}\,\, + \,\,{I_{n - 2}}$ is equal to

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

$n\,\,{\tan ^n}\,\,x\,\, + \,\,c$

The value of $\smallint \,\,{e^{{{\tan }^{ - 1}}x}}\,\,.\,\,\frac{{\left( {1 + x + {x^2}} \right)}}{{1 + {x^2}}}\,\,dx$ is

${e^{{{\tan }^{ - 1}}x}}\,\, + \,\,2x + \,\,c$

If $f\,\,\left( {\frac{{3x\,\, - \,\,4}}{{3x\,\, + \,\,4}}} \right) = x\,\, + \,\,2$, then $\smallint \,\,f\left( x \right)\,\,dx$ is

${e^{x + 2}}\,\,\log \,\,\left| {\frac{{3x\,\, - \,\,4}}{{3x\,\, + \,\,4}}} \right|\,\, + \,\,c$

$\frac{8}{3}\,\,\log \,\,\left| {\,1\,\, - \,\,x\,} \right|\,\, + \,\,\frac{x}{3}\,\, + \,\,c$

${e^{[(3x - 4)/(3x + 4)]}}\,\, - \,\,\frac{{{x^2}}}{2}\,\, - \,\,2x\,\, + \,\,c$

The value of the integral $\smallint \,\,\frac{{\log \,\,\left( {x + 1} \right) - \log \,\,x}}{{x\left( {x + 1} \right)}}\,\,dx$ is

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

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

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

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

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

$\smallint \,\,\frac{{\cos \,\,x - 1}}{{\sin \,\,x + 1}}\,\,{e^x}\,dx$ is equal to

$c\,\, - \,\,\frac{{{e^x}\,\,\sin \,\,x}}{{1 + \,\,\sin \,\,x}}$

$c\,\, - \,\,\frac{{{e^x}}}{{1 + \,\,\sin \,\,x}}$

$c\,\, - \,\,\frac{{{e^x}\,\,\cos \,\,x}}{{1 + \,\,\sin \,\,x}}$

Primitive of ${\cos ^{ - 1}}x\,\,{\rm{w}}.{\rm{r}}.{\rm{t}}\;\,\,x\,\;{\rm{is}}$

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

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

$\smallint \,\,\frac{{10\,\,{x^9} + \,\,{{10}^x}\,\,{{\log }_e}10}}{{{{10}^x} + {x^{10}}}}\,\,dx$ is equal to

${10^x}\,\, - \,\,{x^{10}} + \,\,c$

${10^x}\,\, + \,\,{x^{10}} + \,\,c$

${\left( {{{10}^x}\,\, - \,\,{x^{10}}} \right)^{ - 1}} + \,\,c$

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

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

$\frac{{{e^x}}}{{x\,\, + \,\,1}}\,\, + \,\,c$

$\frac{{x{e^x}}}{{x\,\, + \,\,1}}\,\, + \,\,c$

The value of $\smallint \,\,\frac{{dx}}{{\sqrt x \,\, + \,\,\sqrt[3]{x}}}$ is

$3\,\sqrt x \,\, + \,\,3\,\,\left( {\sqrt[3]{x}} \right)\,\, - \,\,6\,\;\sqrt[6]{x}\,\, + \,\,6\,\,\log \,\,\left( {\sqrt[6]{{x + 1}}} \right)\,\, + \,\,c$

$2\,\sqrt x \,\, + \,\,6\,\,\left( {\sqrt[6]{x}} \right)\,\, - \,\,6\,\,\log \,\,\left( {\sqrt[6]{{x\,\, + \,\,1}}} \right)\,\, + \,\,c$

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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 211-240 of 746 questions

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211 Easy

211 of 746

Application +4 | -1

\int \,[\,\sin \,(\log \,\,x)\,\, + \,\,\cos \,(\log \,\,x)\,]\,dx

is equal to

View answer and explanation

Correct option: C

x\sin \,\,(\log \,\,x) + c

Option C (

x\sin \,(\log \,x) + c

) is correct because the integral can be simplified by recognizing it as the derivative of

x \sin(\log x)

212 Easy

212 of 746

Application +4 | -1

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

for

n \ge 2

, then

{I_n}\,\, + \,\,{I_{n - 2}}

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because integrating by parts and simplifying the expression yields the given result.

213 Easy

213 of 746

Analyzing +4 | -1

\int \,\,\frac{{\sin \,\,x\,\, - \,\,\cos \,\,x}}{{\sqrt {1 - \sin \,\,2x} }}\,\,{e^{\sin x}}\cos \,\,x\,dx

is equal to

View answer and explanation

Correct option: A

{e^{\sin x}} + C

Option A (

{e^{\sin x}} + C

) is correct because by simplifying the integral and using a simple substitution, the integral evaluates to

{e^{\sin x}} + C

214 Easy

214 of 746

Application +4 | -1

\int \frac{1}{(x-1) \sqrt{x^2-1}} d x

equals

View answer and explanation

Correct option: D

-\sqrt{\frac{x+1}{x-1}}+C

Option D is correct because the integral is solved using a substitution method, resulting in the given expression with a square root.

215 Easy

215 of 746

Application +4 | -1

Let

f(x)

be a function satisfying

f^{\prime}(x)=f(x)

and

f(0)=2

. Then

\int \frac{f(x)}{3+4 f(x)} d x

is equal to

View answer and explanation

Correct option: A

(1 / 4) \log \left(3+8 e^x\right)+C

The correct answer is A because solving the differential equation gives

f(x) = 2e^x

, and substituting this into the integral and solving yields the result.

216 Easy

216 of 746

Application +4 | -1

Let

F(x)

be the primitive of

\frac{3 x+2}{\sqrt{x-9}}

w.r.t.

x

. If

F(10)=60

, then the value of

F(13)

View answer and explanation

Correct option: B

132

Option B (132) is correct because it is the value of F(13) after finding the primitive F(x) and using the given condition F(10) = 60 to determine the constant of integration.

217 Easy

217 of 746

Application +4 | -1

The value of

\int \,\,{e^{{{\tan }^{ - 1}}x}}\,\,.\,\,\frac{{\left( {1 + x + {x^2}} \right)}}{{1 + {x^2}}}\,\,dx

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the integral can be solved by recognizing the derivative pattern within the integrand.

218 Easy

218 of 746

Application +4 | -1

\int \sin (\log x) d x

is equal to

View answer and explanation

Correct option: C

\frac{x}{2}[\sin (\ln x)-\cos (\ln x)]+c

Option C is correct because the integral of

\sin(\log x)

simplifies to

\frac{x}{2}[\sin (\ln x)-\cos (\ln x)]+c

after using integration by parts twice.

219 Easy

219 of 746

Application +4 | -1

\int \frac{\cos 2 x}{\cos x} d x

is equal to

View answer and explanation

Correct option: D

2 \sin x-\log |(\sec x+\tan x)|+c

Option D is correct because the integral simplifies to

2\sin x - \log |\sec x + \tan x| + c

after using the trigonometric identity

\cos 2x = 2\cos^2 x - 1

220 Easy

220 of 746

Application +4 | -1

f\,\,\left( {\frac{{3x\,\, - \,\,4}}{{3x\,\, + \,\,4}}} \right) = x\,\, + \,\,2

, then

\int \,\,f\left( x \right)\,\,dx

View answer and explanation

Correct option: B

- \,\,\frac{8}{3}\,\,\log \,\,\left| {\,1\,\, - \,\,x\,} \right|\,\, + \,\,\frac{2}{3}x\,\, + \,\,c

Option B (

\ - \,\frac{8}{3}\\log \left| {\,1\\ - \\,x\} \right|\\ + \\,\frac{2}{3}x\\ + \\,c

) is correct because it results from correct substitution and integration.

221 Easy

221 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of integrating

\tan ^{-1} x

using integration by parts.

222 Easy

222 of 746

Analyzing +4 | -1

The value of the integral

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

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the integral is solved using substitution and properties of logarithms.

223 Easy

223 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

\csc^4 x

by rewriting it as

\csc^2 x (1 + \cot^2 x)

and then integrating.

224 Easy

224 of 746

Application +4 | -1

The value of

\int \frac{\sin x}{\cos ^{3 / 2} x} d x

is equal to

View answer and explanation

Correct option: C

2 \sqrt{\sec x}+c

Option C is correct because substituting

t = \cos x

leads to a simple power rule integration, resulting in

2\sqrt{\sec x} + c

225 Easy

225 of 746

Application +4 | -1

\int {\frac{{dx}}{{\sqrt {1 + x} + \sqrt x }}}

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because rationalizing the denominator and then integrating yields this result.

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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 746

Application +4 | -1

\int \,\,\frac{{\cos \,\,x - 1}}{{\sin \,\,x + 1}}\,\,{e^x}\,dx

is equal to

View answer and explanation

Correct option: A

\frac{{{e^x}\,\,\cos \,\,x}}{{1\,\, + \,\,\sin \,\,x}}\,\, + \,\,c

Option A (

\frac{{{{{e^x}}\cos \\,x}}}}{{{{1\\,\\ + \\,\sin \\,x}}}}\\ + \\,c

) is correct because the integral can be solved by recognizing a pattern similar to

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

227 Easy

227 of 746

Application +4 | -1

Primitive of

{\cos ^{ - 1}}x\,\,\mathrm{w}.\mathrm{r}.\mathrm{t}\;\,\,x\,\;\mathrm{is}

View answer and explanation

Correct option: B

x\,\,{\cos ^{ - 1}}\,\,x\,\, - \,\,\sqrt {1 - {x^2}} + \,\,c

Option B (

x\,{\cos ^{ - 1}}\,x\, - \,\sqrt {1 - {x^2}} + \,c

) is correct because it is obtained by applying integration by parts to the given inverse trigonometric function.

228 Easy

228 of 746

Application +4 | -1

The value of

\int \limits_1^4 \,\,{e^{\sqrt x }}\,\,dx

, is

View answer and explanation

Correct option: B

2\,{e^2}

Option B (

2{e^2}

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

229 Easy

229 of 746

Application +4 | -1

The area bounded by the parabola

y^2=36 x

, the line

\mathrm{x}=1

and

\mathrm{x}

-axis is_____ sq. units.

View answer and explanation

Correct option: A

Option A (2) is correct because the area under the curve y = sin(x) from 0 to pi is calculated by integrating sin(x) from 0 to pi.

230 Easy

230 of 746

Application +4 | -1

\int \,\,\frac{{\sqrt x }}{{x + 1}}\,\,dx = A\,\sqrt x + B\,\,{\tan ^{ - 1}}\,\,\sqrt x + c

, then

View answer and explanation

Correct option: D

A = 2,\,\,B = - 2

Option D is correct because after performing the substitution and integration, the values of A and B are found to be 2 and -2 respectively.

231 Easy

231 of 746

Understanding +4 | -1

\int \,\,\frac{{10\,\,{x^9} + \,\,{{10}^x}\,\,{{\log }_e}10}}{{{{10}^x} + {x^{10}}}}\,\,dx

is equal to

View answer and explanation

Correct option: D

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

Option D is correct because the integral can be solved by recognizing the numerator as the derivative of the denominator, leading to a logarithmic solution.

232 Easy

232 of 746

Application +4 | -1

The probability of the choosing an integer

\lambda

, randomly from the integers

1,2,3, \ldots 2{n}

is proportional to

\log \lambda

. The conditional probability of choosing integer 2, given that an even integer is choosen is

View answer and explanation

Correct option: B

\frac{\log 2}{n \log 2+\log (n !)}

Option B (

\frac{\log 2}{n \log 2+\log (n !)}

) is correct because it is the conditional probability calculated using the given proportionality.

233 Easy

233 of 746

Application +4 | -1

\int \frac{e^x(1+\sin x)}{1+\cos x} \mathrm{~d} x

is equal to

View answer and explanation

Correct option: B

e^x \tan x / 2+C

Option B (

e^x \tan x / 2+C

) is correct because the integral simplifies to the form

\int e^x (f(x) + f'(x)) dx = e^x f(x) + C

, where

f(x) = \tan(x/2)

234 Easy

234 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

{e^x}\,\,\frac{{x\,\, - \,1}}{{x\,\, + \,\,1}}\,\, + \,\,c

Option C (

{e^x}\frac{{x\\ - \\,1}}{{x\\ + \\,1}}\\ + \\,c

) is correct because the integral can be simplified and solved using integration techniques.

235 Easy

235 of 746

Application +4 | -1

The value of

\int \,\,\frac{{dx}}{{\sqrt x \,\, + \,\,\sqrt[3]{x}}}

View answer and explanation

Correct option: C

2\,\sqrt x \,\, - \,\,3\,\,\left( {\sqrt[3]{x}} \right)\,\, + \,\,6\,\;\left( {\sqrt[6]{x}} \right)\;\, - \,6\,\,\log \,\,\left( {\sqrt[6]{{x + 1}}} \right)\,\, + \,\,c

Option C (

2\,\sqrt x \, - \,3\,\left( {\sqrt[3]{x}} \right)\, + \,6\;\left( {\sqrt[6]{x}} \right)\, - \,6\,\log \,\left( {\sqrt[6]{{x + 1}}} \right)\, + \,c

) is correct because it is the result of a u-substitution and polynomial long division.

236 Easy

236 of 746

Application +4 | -1

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

equals

View answer and explanation

Correct option: D

None of these

None of the options are correct. The correct answer is

\frac{2}{\sqrt{7}} \tan^{-1} \frac{(4x+1)}{\sqrt{7}}+C

237 Easy

237 of 746

Application +4 | -1

\int x^2 \sin x d x=

View answer and explanation

Correct option: C

-x^2 \cos x+2 x \sin x+2 \cos x+c

Option C is correct because it is the result of applying integration by parts twice to the integral of

x^2 \sin x

238 Easy

238 of 746

Application +4 | -1

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

equals

View answer and explanation

Correct option: C

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

Option C,

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

, is correct because rewriting the integrand using trigonometric identities and integrating leads to this result.

239 Easy

239 of 746

Application +4 | -1

\int x^2 \sin 2 x d x=

View answer and explanation

Correct option: B

-\frac{1}{2} x^2 \cos 2 x+\frac{1}{2} x \sin 2 x+\frac{1}{4} \cos 2 x+c

Option B (

-\frac{1}{2} x^2 \cos 2 x+\frac{1}{2} x \sin 2 x+\frac{1}{4} \cos 2 x+c

) is correct because it is the result of integrating

x^2 \sin 2x

twice using integration by parts.

240 Easy

240 of 746

Application +4 | -1

The value of

\int \frac{1}{x+\sqrt{x-1}} d x

, is

View answer and explanation

Correct option: C

\log (x+\sqrt{x-1})-\frac{2}{3} \tan ^{-1}\left(\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right)+C

Option C,

\log (x+\sqrt{x-1})-\frac{2}{3} \tan ^{-1}\left(\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right)+C

, is correct because using the substitution

x-1=t^2

and completing the square leads to this result.

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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