$\int \frac{d x}{(x-\beta) \sqrt{(x-\alpha)(\beta-x)}}$ is

$\frac{2}{\alpha-\beta} \sqrt{(x-\alpha)(\beta-x)}+c$

$\frac{\alpha-\beta}{2}(x-\alpha) \sqrt{\beta-x}$

If $x = \mathop \smallint \limits_2^{\sin t} {\sin ^{ - 1}}\,\,\theta \,d\theta $ and $y = \mathop \smallint \limits_n^{\sqrt t } \,\,\frac{{\sin \,\,{\theta ^2}}}{\theta }\,\,d\;\theta $ then $\frac{{dy}}{{dx}}$ is equal to

$\frac{{2\,{t^2}}}{{\tan \,\,t}}$

$\smallint \,\,\frac{{\sqrt x }}{{x\,\, + \,\,1}}\,\,\;dx$ is equal to

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

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

Let $f\left( x \right) = \frac{{{{\sin }^2}\,\,\pi x}}{{1\,\, + \,\,{\pi ^x}}}$. Then , \[\smallint \,\left[ {f\left( x \right)\,\, + \,\,f\left( { - x} \right)} \right]\,\,dx\] is equal to

$\frac{x}{2}\,\, - \,\,\frac{{\cos \,\,\pi x}}{{2\pi }}\,\, + \,\,c$

If $f\left( x \right) = \cos \,\,x\,\, - \,\,{\cos ^2}\,x\,\, + \,\,{\cos ^3}\,\,x\,\, - ...\;\infty $, then $\smallint f\left( x \right)dx$ is equal to

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

$\frac{{x - \tan \,\,\frac{x}{2}}}{2} + c$

$\int \frac{d x}{(x-\beta) \sqrt{(x-\alpha)(\beta-x)}}$ is

$\frac{2}{\alpha-\beta} \sqrt{(x-\alpha)(\beta-x)}+c$

$\frac{\alpha-\beta}{2}({x}-\alpha) \sqrt{\beta-{x}}$

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

$\frac{{{{\left( {x + 1} \right)}^{7/2}}}}{7}\,\, - \,\,2\frac{{{{\left( {x + 1} \right)}^{5/2}}}}{5}\,\, + \,\,5$

$\frac{{{{\left( {x + 1} \right)}^{7/2}}}}{7}\,\, - \,\,3\frac{{{{\left( {x + 1} \right)}^{5/2}}}}{5}\,\, + \,\,11{\left( {x + 1} \right)^{1/2}}\,\, + \,\,c$

If $\smallint \,\,\frac{{\log \,\,\left( {x\,\, + \,\,\sqrt {1\,\, + \,\,{x^2}} } \right)}}{{\sqrt {1\,\, + \,\,{x^2}} }}\;\,\,dx = {\rm{g}}\,\,of\,\,\left( x \right) + $ constant, then

$f\,\,\left( x \right) = \log \,\,\left( {x + \sqrt {{x^2} + \,\,1} } \right)$ and ${\rm{g}}\,\,\left( x \right) = \frac{{{x^2}}}{2}$

$f\,\,\left( x \right) = \log \,\,(x + \sqrt {{x^2} + \,\,1} )$

$f\left( x \right) = \log \,\,\left( {x + \sqrt {{x^2} + \,\,1} } \right)$ and ${\rm{g}}\,\,\left( x \right) = {x^2}$

$f\,\,\left( x \right) = \frac{{{x^2}}}{2}$ and ${\rm{g}}\,\,\left( x \right) = \log \,\,\left( {x\,\, + \,\,\sqrt {{x^2} + \,\,1} } \right)$

$\smallint \,\,{x^3}\,\,\log \,\,x\,dx$ is equal to

$\frac{{{x^4}\,\,\log \,\,x}}{4}\,\, + \,\,c$

$\frac{1}{8}\,\,[{x^4}\,\,\log \,\,x\,\, - \,\,4{x^2}] + c$

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

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

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

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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 211-240 of 411 questions

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211 Medium

211 of 411

Application +4 | -1

\int \frac{d x}{(x-\beta) \sqrt{(x-\alpha)(\beta-x)}}

View answer and explanation

Correct option: A

\frac{2}{\alpha-\beta} \sqrt{\frac{x-\alpha}{\beta-x}}+c

Option A is correct because the integral evaluates to (2/(α-β)) * sqrt((x-α)/(β-x)) + c after using the appropriate substitution and trigonometric identities.

212 Medium

212 of 411

Application +4 | -1

x = \int \limits_2^{\sin t} {\sin ^{ - 1}}\,\,\theta \,d\theta

and

y = \int \limits_n^{\sqrt t } \,\,\frac{{\sin \,\,{\theta ^2}}}{\theta }\,\,d\;\theta

then

\frac{{dy}}{{dx}}

is equal to

View answer and explanation

Correct option: A

\frac{{\tan \,\,t}}{{2\,{t^2}}}

Option A (

\frac{{\tan t}}{{2{t^2}}}

) is correct because it is the derivative dy/dx found using Leibniz's rule and the chain rule.

213 Medium

213 of 411

Application +4 | -1

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

View answer and explanation

Correct option: C

x \log \log x-\frac{x}{\log x}+c

Option C (

x \log \log x - \frac{x}{\log x} + c

) is correct because it is the result of integrating the given expression using integration by parts and substitution.

214 Medium

214 of 411

Application +4 | -1

\int e^x\left[\sin ^{-1} \frac{x}{a}+\frac{1}{\sqrt{a^2-x^2}}\right] d x=

View answer and explanation

Correct option: C

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

Option C is correct because the integral can be expressed in the form

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

, where

f(x) = \sin^{-1}(\frac{x}{a})

and

f'(x) = \frac{1}{\sqrt{a^2 - x^2}}

215 Medium

215 of 411

Application +4 | -1

\int \frac{{{e^x} - \,\,1}}{{{e^x} + \,\,1}}\;dx = f\left( x \right)\,\, + \,\,c

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the result of integrating the given expression after rewriting it.

216 Medium

216 of 411

Application +4 | -1

\int x e^x \cos x d x=(x)+c

, then

f(x)

is equal to

View answer and explanation

Correct option: A

\frac{e^x}{2}\{(1-x) \sin x-x \cos x\}

Option A is correct because the integral can be solved by considering the real part of a complex exponential and using integration by parts.

217 Medium

217 of 411

Application +4 | -1

\int \frac{d x}{\cos (x-a) \cos (x-b)}=

View answer and explanation

Correct option: B

\operatorname{cosec}(a-b) \log \frac{\cos (x-a)}{\cos (x-b)}+c

Option B is correct because using the trigonometric identity

\sin(a-b) = \sin a \cos b - \cos a \sin b

allows us to rewrite the integral and solve it, resulting in

\csc(a-b) \log \frac{\cos(x-a)}{\cos(x-b)} + c

218 Medium

218 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

2\,(\sqrt x - {\tan ^{ - 1}}\,\,\sqrt x ) + c

Option D (

2\,(\sqrt x - {\tan ^{ - 1}}\,\,\sqrt x ) + c

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

219 Medium

219 of 411

Application +4 | -1

\int \cos \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2} x^2+C

Option B (

\frac{{1}}{{2}} x^2+C

) is correct because by using the substitution

x = \cos 2\theta

, the integral simplifies to

\frac{{1}}{{2}}x^2 + C

220 Medium

220 of 411

Application +4 | -1

\int \,\,\frac{{{e^x}(1 + \sin \,\,x)\,\,dx}}{{1 + \cos \,\,x}} = {e^x}f\,\left( x \right)\,\, + \,\,c,\;\,\mathrm{then}\;\,\,f\left( x \right)

is equal to

View answer and explanation

Correct option: C

\tan \,\,\frac{x}{2}

Option C (

\tan \frac{x}{2}

) is correct because it is the function that satisfies the given integral equation after simplification and integration.

221 Medium

221 of 411

Analyzing +4 | -1

Let

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

. Then ,

\int \,\left[ {f\left( x \right)\,\, + \,\,f\left( { - x} \right)} \right]\,\,dx

is equal to

View answer and explanation

Correct option: D

\frac{x}{2}\,\, - \,\,\frac{{\sin \,\,2\pi \,\,x}}{{4\pi }}\,\, + \,\,c

Option D (

\frac{x}{2} - \frac{{\sin 2\pi x}}{{4\pi }} + c

) is correct because it results from simplifying

f(x) + f(-x)

and then integrating the simplified expression.

222 Medium

222 of 411

Analyzing +4 | -1

f\left( x \right) = \cos \,\,x\,\, - \,\,{\cos ^2}\,x\,\, + \,\,{\cos ^3}\,\,x\,\, - ...\;\infty

, then

\int f\left( x \right)dx

is equal to

View answer and explanation

Correct option: B

x - \,\,\tan \,\,\frac{x}{2} + c

Option B (

x - \,\,\tan \,\,\frac{x}{2} + c

) is correct because the given series is a geometric progression that can be simplified and integrated.

223 Medium

223 of 411

Application +4 | -1

The value of

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

is equal to

View answer and explanation

Correct option: C

\frac{x}{1+\ln x}+c

Option C is correct because the integral can be solved by rewriting the integrand and using integration by parts, resulting in

\frac{x}{1+\ln x} + c

224 Medium

224 of 411

Analyzing +4 | -1

\int x^{\frac{13}{2}}\left(1+x^{\frac{5}{2}}\right)^{\frac{1}{2}} d x=a\left(1+x^{\frac{5}{2}}\right)^{\frac{7}{2}}+b\left(1+x^{\frac{5}{2}}\right)^{\frac{5}{2}}+c\left(1+x^{\frac{5}{2}}\right)^{\frac{3}{2}}+d

, then

{b}=

View answer and explanation

Correct option: D

-\frac{8}{25}

Option C (12) is correct because by using vector triple product and given conditions, we can simplify the expression and find its magnitude.

225 Medium

225 of 411

Analyzing +4 | -1

\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|B(\theta)|+C

, where

C

is a constant of integration, then

\frac{B(\theta)}{A}

can be :

View answer and explanation

Correct option: D

\frac{5(2 \sin \theta+1)}{\sin \theta+3}

Option D (

\frac{{5(2 \sin \theta + 1)}}{{(\sin \theta + 3)}}

) is correct because it matches the simplified form of the integral after substitution and partial fraction decomposition.

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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 Medium

226 of 411

Application +4 | -1

\int \frac{d x}{(x-\beta) \sqrt{(x-\alpha)(\beta-x)}}

View answer and explanation

Correct option: A

\frac{2}{\alpha-\beta} \sqrt{\frac{x-\alpha}{\beta-x}}+c

Option B (Only Statement II is true) is correct because the mean is calculated to be 5.16 and the standard deviation is calculated to be 1.54, making only statement II correct.

227 Medium

227 of 411

Application +4 | -1

The integral

\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral can be solved using integration by parts after rewriting the expression.

228 Medium

228 of 411

Application +4 | -1

\int \left( {{x^2} + 1} \right)\,\,\sqrt {x + 1} \;dx

is equal to

View answer and explanation

Correct option: B

2\,\,\left[ {\frac{{{{\left( {x + 1} \right)}^{7/2}}}}{7}\,\, - \,\,2\frac{{{{\left( {x + 1} \right)}^{5/2}}}}{5}\,\, + \,\,2\frac{{{{\left( {x + 1} \right)}^{3/2}}}}{3}} \right] + \,\,c

Option B is correct because the integral is solved using substitution, resulting in the given expression with the correct constant of integration.

229 Medium

229 of 411

Application +4 | -1

\int {\frac{{dx}}{{x\sqrt {{x^4} - 1} }}}

is equal to

View answer and explanation

Correct option: B

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

Option D (

\sin x

) is correct because if

f(x) = \sin x

, then

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

230 Medium

230 of 411

Application +4 | -1

\int \frac{{\sin \,\,x}}{{\cos \,\,x\,\,\left( {1\,\, + \,\,\cos \,\,x} \right)}}\,\,dx = f\left( x \right)\,\, + \,\,c

is equal to

View answer and explanation

Correct option: A

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

Option A is correct because the integral of the given function simplifies to the logarithm of the absolute value of (1 + cos x) / cos x.

231 Medium

231 of 411

Application +4 | -1

\int \,\,\frac{{\log \,\,\left( {x\,\, + \,\,\sqrt {1\,\, + \,\,{x^2}} } \right)}}{{\sqrt {1\,\, + \,\,{x^2}} }}\;\,\,dx = \mathrm{g}\,\,of\,\,\left( x \right) +

constant, then

View answer and explanation

Correct option: C

f\,\,\left( x \right) = \log \,\,\left( {x + \sqrt {{x^2} + \,\,1} } \right)

and

\mathrm{g}\,\,\left( x \right) = \frac{{{x^2}}}{2}

The correct answer is f(x) = log (x + √(x² + 1)) and g(x) = x²/2 because after substitution and integration, the given integral evaluates to this result.

232 Medium

232 of 411

Application +4 | -1

\int \frac{d \theta}{\sin \theta \cos ^3 \theta}=

View answer and explanation

Correct option: C

\log \tan \theta+\frac{1}{2} \tan ^2 \theta+c

Option C (

\log \tan \theta+\frac{1}{2} \tan ^2 \theta+c

) is correct because the integral simplifies to this form after substitution and integration.

233 Medium

233 of 411

Application +4 | -1

\int \frac{x^4\left(x^{10}-1\right)}{x^{20}+3 x^{10}+1} d x

equals

View answer and explanation

Correct option: B

\frac{1}{5}{\operatorname{Tan} ^{ - 1}}\left( {{x^5} + \frac{1}{{{x^5}}}} \right) + c

Option B (

\frac{{1}}{{5}}{{\operatorname{{Tan}} ^{ - 1}}}\left( {{{{x^5} + \frac{{1}}{{{{x^5}}}}}}} \right) + c

) is correct because it is the result of the integration after substitution and simplification.

234 Medium

234 of 411

Application +4 | -1

\int \frac{x^3-3 x+7}{x^2+4} d x

is equal to

View answer and explanation

Correct option: B

\frac{x^2}{2}+\frac{7}{2} \tan ^{-1} \frac{x}{2}-\frac{7}{2} \ln \left(x^2+4\right)+c

Option B (

\frac{x^2}{2}+\frac{7}{2} \tan ^{-1} \frac{x}{2}-\frac{7}{2} \ln \left(x^2+4\right)+c

) is correct because after polynomial long division and separation of the integral, the result matches this option.

235 Medium

235 of 411

Application +4 | -1

\int \,\,{x^3}\,\,\log \,\,x\,dx

is equal to

View answer and explanation

Correct option: B

\frac{1}{{16}}\,\,[4{x^4}\,\,\log \,\,x\,\, - \,\,{x^4}] + c

Option B (

\frac{{1}}{{16}}\left[ {4{x^4}\log x - {x^4}} \right] + c

) is correct because it is the result of integrating the given expression using integration by parts.

236 Medium

236 of 411

Understanding +4 | -1

The value of

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

is equal to

View answer and explanation

Correct option: A

x + c

Option A (x + c) is correct because the integral simplifies to arcsin(sin(x)) + c, which equals x + c.

237 Medium

237 of 411

Analyzing +4 | -1

The value of

\int \frac{d x}{\cos ^6 x+\sin ^6 x}

, is equal to

View answer and explanation

Correct option: D

\tan ^{-1}(-2 \cot 2 x)+C

Option D (

\tan^{-1}(-2 \cot 2x) + C

) is correct because the integral simplifies to this form after applying trigonometric identities and substitution.

238 Medium

238 of 411

Application +4 | -1

\int \frac{1}{x^3}\left[\log x^x\right]^2 d x=

View answer and explanation

Correct option: B

\frac{1}{3}(\log x)^3+c

Option B (

\frac{{1}}{{3}}(\log x)^3+c

) is correct because the integral simplifies to this form after using properties of logarithms and substitution.

239 Medium

239 of 411

Analyzing +4 | -1

\int \frac{\operatorname{cosec}^2 x-2005}{\cos ^{2005} x} d x

is equal to

View answer and explanation

Correct option: D

-\frac{\cot x}{(\cos x)^{2005}}+c

Option C (

y=\cot ^{-1} x

) is correct because it is derived by taking the limit as h approaches 0 and solving for y.

240 Medium

240 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

x\,\;{e^{{{\tan }^{ - 1}}x}}\,\, + \,\,C

Option A (

x\;{e^{{{\tan }^{ - 1}}x}}\, + \,C

) is correct because it results from a u-substitution and simplification of the integral.

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