If $\smallint \,\,\sqrt {\frac{x}{{{a^3} - {x^3}}}} \,\,dx = {\rm{g}}\left( x \right) + c$, then g(x) is equal to

$\frac{2}{3}\,\,{\cos ^{ - 1}}\,\,x$

$\frac{2}{3}\,\,{\sin ^{ - 1}}\,\,\left( {\frac{{{x^3}}}{{{a^3}}}} \right)$

$\frac{2}{3}\,\,{\cos ^{ - 1}}\,\,\left( {\frac{x}{a}} \right)$

Let $\smallint \,\,\sqrt {\frac{{5\,\, - \,\,x}}{{2\,\, + \,\,x}}} \,\,dx$ equal

$\sqrt {x + 2} \,\,\sqrt {5 - x} \,\, + \,\,3\,\,{\sin ^{ - 1}}\,\,\sqrt {\frac{{x + 2}}{3}} + C$

$\sqrt {x + 2} \,\,\sqrt {5 - x} \,\, + \,\,5\,\,{\sin ^{ - 1}}\,\,\sqrt {\frac{{x + 2}}{5}} + C$

$\smallint \,\,\frac{{{x^4} + 1}}{{{{\left( {1 - {x^4}} \right)}^{3/2}}}}\,\,dx$ is equal to

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

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

The value of $\mathop \smallint \nolimits_1^2 \,\,\frac{{dx}}{{x\left( {1 + {x^4}} \right)}}$ is

$\frac{1}{4}\,\,\log \,\,\frac{{17}}{{32}}$

$\frac{1}{4}\,\,\log \,\,\frac{{17}}{2}$

$\smallint \frac{{x + 2}}{{\left( {{x^2} + 3x + 3} \right)\,\,\sqrt {x + 1} }}dx$ is equal to

$\frac{1}{{\sqrt 3 }}\,\,{\tan ^{ - 1}}\,\,\left( {\frac{x}{{\sqrt {3\left( {x + 1} \right)} }}} \right)$

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

If $f(x)=\int \frac{\tan ^3 x}{2+\tan ^2 x} d x$, then

$f(x)=\log \left|\frac{2-\cos ^2 x}{\sin x}\right|+C$

$f(x)=\log \left|\frac{2+\cos ^2 x}{\sin x}\right|+C$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

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181 Medium

181 of 411

Application +4 | -1

\int \,\,\sqrt {\frac{x}{{{a^3} - {x^3}}}} \,\,dx = \mathrm{g}\left( x \right) + c

, then g(x) is equal to

View answer and explanation

Correct option: C

\frac{2}{3}\,\,{\sin ^{ - 1}}\,\,\left( {\sqrt {\frac{{{x^3}}}{{{a^3}}}} } \right)

Option C is correct because the integral can be solved using trigonometric substitution, leading to an inverse sine function.

182 Medium

182 of 411

Application +4 | -1

\int \frac{{\sin \,\,x}}{{\sin \,\,\left( {x - \alpha } \right)}}\,\,dx = Ax + B\,\,{\log _e}\,\,\sin \,\left( {x - \alpha } \right) + C

, then

View answer and explanation

Correct option: C

A = \cos \,\,\alpha ,\;\,B = \sin \,\,\alpha

Option C is correct because by expanding

\sin x

\sin((x-\alpha) + \alpha)

and integrating, we find

A = \cos \alpha

and

B = \sin \alpha

183 Medium

183 of 411

Application +4 | -1

\int x^{-6}\left(1+2 x^3\right)^{2 / 3} d x

equals

View answer and explanation

Correct option: A

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

Option A (

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

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

184 Medium

184 of 411

Application +4 | -1

Let

\int \,\,\sqrt {\frac{{5\,\, - \,\,x}}{{2\,\, + \,\,x}}} \,\,dx

equal

View answer and explanation

Correct option: B

\sqrt {x + 2} \,\,\sqrt {5 - x} \,\, + \,\,7\,\,{\sin ^{ - 1}}\,\,\sqrt {\frac{{x + 2}}{7}} + C

Option B (

\sqrt {x + 2} \,\sqrt {5 - x} \, + \,7\,{\sin ^{ - 1}}\,\sqrt {\frac{{x + 2}}{7}} \, + C

) is correct because after substituting

2 + x = t^2

, the integral can be solved using standard integral formulas.

185 Medium

185 of 411

Application +4 | -1

The primitive function of the function

f\,\,\left( x \right) = \,\,\frac{{\sqrt {\left( {{a^2} - {x^2}} \right)} }}{{{x^4}}}

View answer and explanation

Correct option: C

c\,\, - \,\,\frac{{{{\left( {{a^2} - {x^2}} \right)}^{3/2}}}}{{3{a^2}\,{x^3}}}

Option C (

c\\,\\ - \\,\frac{{{{\left( {{a^2} - {x^2}} \right)}^{3/2}}}}{{3{a^2}\,{x^3}}}

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

186 Medium

186 of 411

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: D

\frac{1}{\sqrt{2}} \operatorname{Cos}^{-1} \frac{x \sqrt{2}}{x^2+1}+c

Option D is correct because the integral simplifies to an inverse cosine form after dividing by x² and using a substitution.

187 Medium

187 of 411

Application +4 | -1

\int \frac{x \cos x}{(x \sin x+\cos x)^2} d x

is equal to

View answer and explanation

Correct option: B

-\frac{1}{(x \sin x+\cos x)}+k

Option B (

\frac{-1}{(x \sin x+\cos x)}+k

) is correct because the integral simplifies to this form after recognizing the derivative in the numerator.

188 Medium

188 of 411

Application +4 | -1

\int \cos \left(\log _e x\right) d x

is equal to

View answer and explanation

Correct option: C

\frac{x}{2}\left[\cos \left(\log _e x\right)+\sin \left(\log _e x\right)\right]+c

Option C (

\frac{x}{2}\left[\cos \left(\log _e x\right)+\sin \left(\log _e x\right)\right]+c

) is correct because by using integration by parts twice, we can solve the integral.

189 Medium

189 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the integral can be solved by rewriting the expression and using a suitable substitution.

190 Medium

190 of 411

Application +4 | -1

\int \,\,\left| {\,x\,} \right|\,\,\log \,\,\left| {\,x\,} \right|\,\,dx

equals

\left( {x \ne 0} \right)

View answer and explanation

Correct option: D

\frac{1}{2}x\,\,\left| {\,x\,} \right|\,\,\log \,\,\,\left| {\,x\,} \right|\,\, - \,\,\frac{1}{4}\,\,x\,\,\left| {\,x\,} \right|\,\, + \,\,c

Option D (

\frac{1}{2}x\,\left| {\,x\} \right|\,\log \,\,\left| {\,x\} \right|\, - \,\frac{1}{4}\,x\,\left| {\,x\} \right|\, + \,c

) is correct because it is the result of integrating |x|log|x| using integration by parts, considering both positive and negative cases for x.

191 Medium

191 of 411

Application +4 | -1

\int \frac{\left(2 x^{12}+5 x^9\right) d x}{\left(1+x^3+x^5\right)^3}=

View answer and explanation

Correct option: A

\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+c

Option A (

\frac{{x^{{10}}}}{{2(x^5+x^3+1)^2}}+c

) is correct because after substitution and integration, the result matches this option.

192 Medium

192 of 411

Application +4 | -1

\int \frac{x^2-1}{\left(x^2+1\right) \sqrt{x^4+1}} d x=

View answer and explanation

Correct option: B

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

Option B (

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

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

193 Medium

193 of 411

Application +4 | -1

\int \frac{\sqrt{x}}{\sqrt{a^3-x^3}} d x=

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because by substituting

x=a \sin ^{2 / 3} \theta

, we can evaluate the integral.

194 Medium

194 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

2 \sqrt{x^2+2}+C

Option A (

2 \sqrt{{x^2+2}}+C

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

195 Medium

195 of 411

Analyzing +4 | -1

\int \frac{d x}{\left(x^2+a^2\right)^3}=\frac{x}{4 a^2\left(x^2+a^2\right)}+\frac{m}{n a^2}

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

. Then

|m-n|

is equal to

View answer and explanation

Correct option: D

Option D is correct because after evaluating the integral and comparing coefficients, the absolute difference between m and n is found to be 1.

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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

196 Medium

196 of 411

Analyzing +4 | -1

Let

f\left( x \right) = \frac{x}{{{{\left( {1 + {x^n}} \right)}^{1/n}}}}

for

n\,\, \ge 2

and

\mathrm{g}\,\,\left( x \right) = \left( {fofo\,\, \ldots of} \right)\,\,\left( x \right)

g(x) = \underbrace {(fofo....of)}_{n\,\,times}\,\,(x)

Then,

\int \,\,{x^{n - 2}}\,\,\mathrm{g}\left( x \right)\,\,dx

equals

View answer and explanation

Correct option: A

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

Option A is correct because by finding the composite function g(x) and then integrating x^(n-2)*g(x), we arrive at the given expression.

197 Medium

197 of 411

Analyzing +4 | -1

\int \frac{\sin ^4 x}{\cos x} d x=\frac{1}{2} \log _e\left(\frac{1+\sin x}{1-\sin x}\right)-g(x)+c

where

g(x)

equals

View answer and explanation

Correct option: A

\frac{1}{3} \sin ^3 x+\sin x

Option A (

\left[-1,-\frac{3}{4}\right]

) is correct because the range of the expression

\frac{1}{4}(\sin 2 \theta)^2-1

falls within this interval.

198 Medium

198 of 411

Application +4 | -1

The value of

\int \nolimits_1^2 \,\,\frac{{dx}}{{x\left( {1 + {x^4}} \right)}}

View answer and explanation

Correct option: B

\frac{1}{4}\,\,\log \,\,\frac{{32}}{{17}}

Option B (

\frac{1}{4}\,\log \,\frac{{32}}{{17}}

) is correct because after substituting

x^4 = t

and using partial fractions, we can evaluate the definite integral.

199 Medium

199 of 411

Application +4 | -1

\int \frac{2 x+5}{\sqrt{7-6 x-x^2}} d x=A \sqrt{7-6 x-x^2}+B \sin ^{-1}\left(\frac{x+3}{4}\right)+C

(where

C

is a constant of integration), then the ordered pair

(A, B)

is equal to

View answer and explanation

Correct option: A

(-2,-1)

Option A ((-2,-1)) is correct because it is the result of integrating the given function and comparing the coefficients with the given form.

200 Medium

200 of 411

Application +4 | -1

\int \frac{{x + 2}}{{\left( {{x^2} + 3x + 3} \right)\,\,\sqrt {x + 1} }}dx

is equal to

View answer and explanation

Correct option: B

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

Option B,

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

, is correct because it is the result of the given integral after applying a suitable substitution and trigonometric simplification.

201 Medium

201 of 411

Application +4 | -1

\int \log 2 x d x

View answer and explanation

Correct option: D

x \log 2 x-x+c

Option D (

x \log 2 x-x+c

) is correct because it is the result of integrating

\log 2x

using integration by parts.

202 Medium

202 of 411

Analyzing +4 | -1

For

0 < x < 1

, let

f(x)=\lim \limits_{{n \rightarrow \infty}}(1+x)\left(1+x^2\right)\left(1+x^4\right) \cdots\left(1+x^{2^n}\right)

then

\int \frac{f(x)}{1-x} \log _e x d x

equals

View answer and explanation

Correct option: B

-\log _e\left(\frac{x}{1-x}\right)+\frac{\log _e x}{1-x}+c

Option B is correct because the integral can be solved by simplifying the limit, then using integration by parts.

203 Medium

203 of 411

Analyzing +4 | -1

f(x)=\int \frac{\tan ^3 x}{2+\tan ^2 x} d x

, then

View answer and explanation

Correct option: B

f(x)=\log \left|\frac{2-\sin ^2 x}{\cos x}\right|+C

The correct answer is

f(x)=\log \left|\frac{{2-\sin ^2 x}}{{\cos x}}\right|+C

(Option B) because the integral simplifies to this expression after trigonometric substitution and simplification.

204 Medium

204 of 411

Application +4 | -1

\int \frac{{dx}}{{\sin \,\,x\,\,\cos \,\,x}}

is equal to

View answer and explanation

Correct option: B

\log \,\,|\tan \,\,x| + \,c

Option B is correct because the integral of

\frac{{dx}}{{\sin \,x\,\cos \,x}}

simplifies to

\log \,|\tan \,x| + \,c

using trigonometric identities and integration rules.

205 Medium

205 of 411

Application +4 | -1

\int \frac{x^5}{\sqrt{1+x^3}} d x=

View answer and explanation

Correct option: C

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

Option C is correct because the integral can be solved by substituting

1+x^3 = t^2

and simplifying the resulting expression.

206 Medium

206 of 411

Application +4 | -1

\int \frac{3+2 \cos x}{(2+3 \cos x)^{2}} d x

is equal to

View answer and explanation

Correct option: A

\left(\frac{\sin x}{2+3 \cos x}\right)+c

Option A (

\left(\frac{{\sin x}}{{2+3 \cos x}}\right)+c

) is correct because differentiating this expression results in the integrand.

207 Medium

207 of 411

Application +4 | -1

The value of

\int \frac{\log (x / e)}{(\log x)^2} d x

is equal to

View answer and explanation

Correct option: C

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

208 Medium

208 of 411

Application +4 | -1

\mathrm{P}=\int \mathrm{e}^{\mathrm{ax}} \cos \mathrm{bx} d \mathrm{x}

and

\mathrm{Q}=\int \mathrm{e}^{\mathrm{ax}} \sin \mathrm{bx} d \mathrm{x}

, then

\left(\mathrm{P}^2+\mathrm{Q}^2\right)

is equal to (neglect integration constant)

View answer and explanation

Correct option: C

\frac{e^{2 a x}}{a^2+b^2}

Option C (

\frac{e^{2 a x}}{a^2+b^2}

) is correct because by using complex numbers and taking the modulus, the result matches this option.

209 Medium

209 of 411

Application +4 | -1

Let

f(x)=x+\sin x

. Suppose

g

denotes the inverse function of

f

. The value of

g^{\prime}\left(\frac{\pi}{4}+\frac{1}{\sqrt{2}}\right)

has the value equal to

View answer and explanation

Correct option: C

2-\sqrt{2}

Option C (

2-\sqrt{2}

) is correct because it is the value of the derivative of the inverse function

g

evaluated at the given point, calculated using the derivative of the original function

f

210 Medium

210 of 411

Application +4 | -1

\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x}) d x

View answer and explanation

Correct option: A

2 e^{\sqrt{x}}[x-\sqrt{x}+1]+C

Option A (

2 e^{\sqrt{x}}[x-\sqrt{x}+1]+C

) is correct because after substituting

x=t^2

and integrating by parts twice, we obtain the given 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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