If $I=\int \sec ^{-1} \sqrt{x} d x$, then $I$ equals

$x \mathrm{sec}^{-1} \sqrt{x}-\log (1+x)+C$

$\sec ^{-1} \sqrt{x}-\tan ^{-1} \sqrt{x}+C$

$x \sec ^{-1} \sqrt{x}+\log (1+x)+C$

$\smallint \,\,{\cos ^{ - 1}}\,\,\left( {\frac{1}{x}} \right)\,\,dx$ equals

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

Let $f$ be a non-negative function defined on the interval $\left[ {0,1} \right]$. If $\mathop \smallint \nolimits_0^x \,\,\sqrt {1\,\, - \,\,{{\left( {\,f'\left( t \right)} \right)}^2}} \,\,dt = \mathop \smallint \nolimits_0^x \,\,f\,\,\left( t \right)\,\,dt,\,\,0 \le \,\,x\,\, \le 1$ and $f\,\,\left( 0 \right) = 0$, then

$f\,\,\left( {\frac{1}{2}} \right)\,\, \,\,\frac{1}{3}$

$\smallint \frac{{dx}}{{x\,\sqrt {{x^6}\,\, - \,\,16} }}$ is equal to

$\frac{1}{3}\,\,{\sec ^{ - 1}}\left( {\frac{{{x^3}}}{4}} \right)\,\, + \,\,c$

${\cos ^{ - 1}}\,\,\left( {\frac{{{x^3}}}{4}} \right)\,\, + \,\,c$

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

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

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

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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 271-300 of 995 questions

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271 Easy

271 of 995

Application +4 | -1

\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x

is equal to

View answer and explanation

Correct option: D

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

Option D is correct because by substituting

z = 2 - \frac{2}{x^2} + \frac{1}{x^4}

, the integral simplifies to

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

272 Medium

272 of 995

Application +4 | -1

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

, then

\frac{d}{{dx}}\,\,\left\{ {f\left( {{x^3}} \right)} \right\}

View answer and explanation

Correct option: B

\frac{{3{x^2}}}{{1\,\, + \,\,{x^6}}}

Option B (

\frac{{3{x^2}}}{{1\\,\\ + \\,{x^6}}}

) is correct because it is the result of applying the chain rule after recognizing that

f(x) = \tan^{-1}(x)

273 Easy

273 of 995

Application +4 | -1

I=\int \sec ^{-1} \sqrt{x} d x

, then

I

equals

View answer and explanation

Correct option: C

x \sec ^{-1} \sqrt{x}-\sqrt{x-1}+C

Option C (\

x \sec^{-1} \sqrt{x} - \sqrt{x-1} + C

is correct. The integral is solved by using substitution and integration by parts.

274 Easy

274 of 995

Analyzing +4 | -1

Let

f(x)=\left\{\sin \left(\tan ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right\}^2-1

, where

|x|>1

and

\frac{d y}{d x}=\frac{1}{2} \frac{d}{d x}\left(\sin ^{-1} f(x)\right)

. If

y(\sqrt{3})=\frac{\pi}{3}

, then

y(-\sqrt{3})=

View answer and explanation

Correct option: B

-\frac{\pi}{3}

Option B (-

\frac{\pi}{3}

) is correct because after simplifying f(x) and using the given condition, we find y(-

\sqrt{3}

) = -

\frac{\pi}{3}

275 Easy

275 of 995

Application +4 | -1

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

equals

View answer and explanation

Correct option: B

x\,\,{\sec ^{ - 1}}\,\,x\,\, - \,\,{\cosh ^{ - 1}}\,\,\,x\,\, + \,\,c

Option B (

x\,{\sec ^{ - 1}}\,x\, - \,{\cosh ^{ - 1}}\,x\, + \,c

) is correct because it is the result of integration by parts and simplification using inverse hyperbolic functions.

276 Easy

276 of 995

Application +4 | -1

\int(\sin 2 x-\cos 2 x) \mathrm{d} x=\frac{1}{\sqrt{2}} \sin (2 x+k)+C

then

View answer and explanation

Correct option: B

k=5 \pi / 4

Option B is correct because after integrating by parts and simplifying, the integral evaluates to

(x-1) \log (1-\sqrt{x})-\frac{1}{2} x-\sqrt{x}+C

277 Easy

277 of 995

Analyzing +4 | -1

Let

f

be a non-negative function defined on the interval

\left[ {0,1} \right]

. If

\int \nolimits_0^x \,\,\sqrt {1\,\, - \,\,{{\left( {\,f'\left( t \right)} \right)}^2}} \,\,dt = \int \nolimits_0^x \,\,f\,\,\left( t \right)\,\,dt,\,\,0 \le \,\,x\,\, \le 1

and

f\,\,\left( 0 \right) = 0

, then

View answer and explanation

Correct option: C

f\,\,\left( {\frac{1}{2}} \right)\,\, < \,\,\frac{1}{2}

and

f\,\,\left( {\frac{1}{3}} \right)\,\, < \,\,\frac{1}{3}

Option C (

f\,\,\left( {\frac{1}{2}} \right)\,\,< \,\,\frac{1}{2}

and

f\,\,\left( {\frac{1}{3}} \right)\,\,< \,\,\frac{1}{3}

) is correct because

f(x) = \sin(x)

and

\sin(x) < x

for all positive x.

278 Easy

278 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\frac{{{x^3}}}{3}\,\, + \,\,\frac{{{x^2}}}{2}\,\, + \,\,x\,\, + \,\,c

Option B (

\frac{{{x^3}}}{3}\, + \,\,\frac{{{x^2}}}{2}\, + \,x\, + \,c

) is correct because the integral is solved by polynomial long division and then integrating the resulting polynomial.

279 Medium

279 of 995

Application +4 | -1

\int \frac{{dx}}{{x\,\sqrt {{x^6}\,\, - \,\,16} }}

is equal to

View answer and explanation

Correct option: C

\frac{1}{{12}}\,\,{\sec ^{ - 1}}\,\,\left( {\frac{{{x^3}}}{4}} \right)\,\, + \,\,c

Option C (

\frac{1}{{12}}\,{\sec ^{ - 1}}\,\,\left( {\frac{{{x^3}}}{4}} \right)\,\,+\,c

) is correct because the integral is solved using substitution and the standard integral of

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

280 Easy

280 of 995

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)

281 Medium

281 of 995

Analyzing +4 | -1

f: \mathbb{R} \rightarrow \mathbb{R}

is a function satisfying the following:
(i)

f(-x)=-f(x)

(ii)

f(x+1)=f(x)+1

(iii)

f\left(\frac{1}{x}\right)=\frac{f(x)}{x^2} \forall x \neq 0

then

\int e^x f(x) d x

is equal to

View answer and explanation

Correct option: A

e^x(x-1)+c

The correct answer is A (

e^x(x-1)+c

) because after determining that

f(x) = x

, the integral

\int e^x f(x) dx

simplifies to

\int xe^x dx

, which can be solved using integration by parts.

282 Easy

282 of 995

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.

283 Easy

283 of 995

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

284 Easy

284 of 995

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.

285 Easy

285 of 995

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.

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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

286 Easy

286 of 995

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.

287 Easy

287 of 995

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.

288 Medium

288 of 995

Analyzing +4 | -1

\int \frac{x^2 \cos x}{(1+\sin x)^2} d x= \frac{-x^2}{1+\sin x}-2 x(\sec x-\tan x)+2 \log |f(x)|+c

where

f(x)

is equal to

View answer and explanation

Correct option: C

1+\sin x

Option C (

1+\sin x

) is correct because, after integration by parts and simplification, the logarithmic term involves

1+\sin x

289 Easy

289 of 995

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.

290 Easy

290 of 995

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.

291 Easy

291 of 995

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.

292 Medium

292 of 995

Application +4 | -1

\int \frac{d x}{\sqrt[3]{x^{5 / 2}(x+1)^{7 / 2}}}

is equal to:

View answer and explanation

Correct option: B

6\left(\frac{x+1}{x}\right)^{-1 / 6}+C

Option B (

6\left(\frac{x+1}{x}\right)^{-1 / 6}+C

) is correct because it results from the correct substitution and integration of the given expression.

293 Easy

293 of 995

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.

294 Easy

294 of 995

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.

295 Easy

295 of 995

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

296 Easy

296 of 995

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.

297 Medium

297 of 995

Analyzing +4 | -1

\int \frac{d x}{\cos (2 x) \cos (4 x)}

is equal to

View answer and explanation

Correct option: A

\frac{1}{2 \sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1-\sqrt{2} \sin 2 x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C

Option A (

\frac{1}{2 \sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1-\sqrt{2} \sin 2 x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C

) is correct because it is the result of evaluating the integral using trigonometric identities and substitution.

298 Medium

298 of 995

Application +4 | -1

\int \frac{\sqrt{1+x^2}}{x^4} d x

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

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

299 Easy

299 of 995

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.

300 Medium

300 of 995

Analyzing +4 | -1

m

is a non-zero number and

\int \frac{x^{5 m-1}+2 x^{4 m-1}}{\left(x^{2 m}+x^m+1\right)^3} d x=f(x)+c

, then

f(x)

View answer and explanation

Correct option: B

\frac{x^{4 m}}{2 m\left(x^{2 m}+x^m+1\right)^2}

Option B (

\frac{x^{4 m}}{2 m\left(x^{2 m}+x^m+1\right)^2}

) is correct because it is the result of integrating the given expression after substitution and simplification.

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