If an anti-derivate of $f\,\,\left( x \right)$ is e^x and that of ${\rm{g}}\,\,\left( x \right)$ is cos⁡x, then $\smallint \,\,f\left( x \right)\,\,\cos \,\,x\,\,dx\,\, + \,\,\smallint \,\,{\rm{g}}\,\,\left( x \right)\,\,{e^x}\,dx$ is equal to

$f\,\,\left( x \right){\rm{g}}\,\,\left( x \right)\,\, + \,\,c$

$f\,\,\left( x \right)\,\, + \,\,{\rm{g}}\,\,\left( x \right)\,\, + \,\,c$

$f\,\,\left( x \right)\,\, - \,\,{\rm{g}}\,\,\left( x \right)\,\, + \,\,c$

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

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

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

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

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

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

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

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

Assuming that $f$ is everywhere continuous, $\frac{1}{c}\,\,\mathop \smallint \limits_{ac}^{bc} \,\,f\,\,\left( {\frac{x}{c}} \right)\,\,dx$ is equal to

$\mathop \smallint \limits_a^b \,\,f\left( x \right)\,\,dx$

$\frac{1}{c}\,\,\mathop \smallint \limits_a^b \,\,f\left( x \right)\,\,dx$

$c\,\mathop \smallint \limits_a^b \,\,f\left( x \right)\,\,dx$

$\mathop \smallint \limits_{a{c^2}}^{b{c^2}} \,f\,\left( x \right)\,\,dx$

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

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

\[\smallint \,\,\frac{{{a^{x/2}}}}{{\sqrt {{a^{ - x}} - \,\,{a^x}} }}\,\,dx\] is equal to

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

$2\,\,\sqrt {{a^{ - x}} - {a^x}} \,\, + \,\,c$

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

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

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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511 Easy

511 of 995

Application +4 | -1

{I}=\int \frac{d x}{x^3\left(x^8+1\right)^{3 / 4}}=\frac{\lambda\left(1+x^8\right)^{\frac{1}{4}}}{x^2}+c

(where

c

is the constant of integration), then the value of

\lambda

is equal to

View answer and explanation

Correct option: D

-\frac{1}{2}

Option D (-

\frac{{1}}{{2}}

) is correct because it is the value of

\lambda

obtained after solving the integral.

512 Easy

512 of 995

Application +4 | -1

\int x^5 \sqrt[3]{\left(1+x^3\right)^2} d x

is equal to

View answer and explanation

Correct option: A

\frac{1}{8} u^{8 / 3}-\frac{1}{5} u^{5 / 3}+C, u=1+x^3

Option A (

\frac{1}{8} u^{8 / 3}-\frac{1}{5} u^{5 / 3}+C, u=1+x^3

) is correct because after the substitution

u = 1 + x^3

, the integral simplifies and can be easily integrated.

513 Easy

513 of 995

Application +4 | -1

If an anti-derivate of

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

is e^x and that of

\mathrm{g}\,\,\left( x \right)

is cos⁡x, then

\int \,\,f\left( x \right)\,\,\cos \,\,x\,\,dx\,\, + \,\,\int \,\,\mathrm{g}\,\,\left( x \right)\,\,{e^x}\,dx

is equal to

View answer and explanation

Correct option: C

{e^x}\,\,\cos \,\,x\,\, + \,\,c

Option C (

{e^x}\,\,\cos \,\,x\,\, + \,\,c

) is correct because integrating by parts and simplifying leads to this result.

514 Easy

514 of 995

Application +4 | -1

The integral

\int {\cos \left( {{{\log }_e}x} \right)dx}

is equal to (where C is a constant of integration)

View answer and explanation

Correct option: A

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

Option A is correct. The integral is solved using integration by parts twice, leading to the expression (x/2)[cos(log_e(x)) + sin(log_e(x))] + C.

515 Easy

515 of 995

Application +4 | -1

\int \frac{\mathrm{d} x}{\sqrt{2-3 x-x^2}}=f o g(x)+C

then

View answer and explanation

Correct option: C

f(x)=\sin ^{-1} x, g(x)=\frac{2 x+3}{\sqrt{17}}

Option C (

f(x)=\sin ^{-1} x, g(x)=\frac{{2 x+3}}{{\sqrt{{17}}}}

) is correct because completing the square and using the substitution

u = \frac{{2x+3}}{{\sqrt{{17}}}}

leads to the arcsine integral.

516 Easy

516 of 995

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

\frac{-x^2}{(x \tan x+1)}+2 \ln |x \sin x+\cos x|+C

Option B (

\frac{-x^2}{(x \tan x+1)}+2 \ln |x \sin x+\cos x|+C

) is correct because integrating by parts and simplifying leads to this result.

517 Easy

517 of 995

Analyzing +4 | -1

\int \frac{e^{\left(x^2+4 \ln x\right)}-x^3 e^{x^2}}{x-1} d x

is equal to

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to

\frac{1}{2}(x^2-1) e^{x^2}+C

, which is not among the given options.

518 Easy

518 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\sin x+\cos x+c

Option C (

\sin x+\cos x+c

) is correct because the integral simplifies to

\int (\cos x - \sin x) dx

, which results in

\sin x + \cos x + c

519 Medium

519 of 995

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.

520 Easy

520 of 995

Analyzing +4 | -1

\int \frac{\sin ^3 x}{\left(\cos ^4 x+3 \cos ^2 x+1\right) \tan ^{-1}(\sec x+\cos x)} d x

is equal to

View answer and explanation

Correct option: B

\log _e\left|\tan ^{-1}(\sec x+\cos x)\right|+C

Option B is correct because the integral simplifies to a logarithmic form involving the inverse tangent function.

521 Easy

521 of 995

Analyzing +4 | -1

Let

\int \frac{\cos x}{\sin ^3 x\left(1+\sin ^6 x\right)^{2 / 3}} d x=f(x)\left(1+\sin ^6 x\right)^{1 / \lambda}+C.

Then

\lambda f\left(\frac{\pi}{3}\right)=

View answer and explanation

Correct option: B

-2

Option B (-2) is correct because after simplifying and substituting, the value of

\lambda f(\frac{\pi}{3})

is -2.

522 Easy

522 of 995

Application +4 | -1

\int \,\,\frac{1}{{\sin \,\,\left( {x - a} \right)\,\,\cos \,\,\left( {x - b} \right)}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{1}{{\cos \,\,\left( {a - b} \right)}}\,\,\log \,\,\left| {\,\frac{{\sin \,\,\left( {x - a} \right)}}{{\cos \,\,\left( {x - b} \right)}}\,} \right| + C

Option B is correct because the integral is solved by manipulating the integrand using trigonometric identities and then integrating.

523 Easy

523 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

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

524 Medium

524 of 995

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}}

525 Medium

525 of 995

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.

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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

526 Medium

526 of 995

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.

527 Easy

527 of 995

Application +4 | -1

f(x)=\sqrt{4 x^2+4 x-3}

then

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

is equal

View answer and explanation

Correct option: B

(1 / 4) f(x)+(5 / 4) \log |2 x+1+f(x)|+C

The correct answer is B because after substituting

2x+1=t

and integrating, we obtain the given expression.

528 Easy

528 of 995

Application +4 | -1

Assuming that

f

is everywhere continuous,

\frac{1}{c}\,\,\int \limits_{ac}^{bc} \,\,f\,\,\left( {\frac{x}{c}} \right)\,\,dx

is equal to

View answer and explanation

Correct option: B

\int \limits_a^b \,\,f\left( x \right)\,\,dx

Option B (

\int \limits_a^b \,f(x)\,dx

) is correct because after substitution and simplification, the integral transforms to this form.

529 Medium

529 of 995

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

530 Medium

530 of 995

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.

531 Medium

531 of 995

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.

532 Hard

532 of 995

Understanding +4 | -1

\ell^{\mathrm{r}}

means

(\log \log \log \ldots.) \mathrm{x}

, the

\log

being repeated

\mathrm{r}

times, then

\int\left\{\mathrm{x} \ell(\mathrm{x}) \ell^2(\mathrm{x}) \ell^3(\mathrm{x}) \ldots \ell^{\mathrm{r}}(\mathrm{x})\right\}^{-1} \mathrm{dx}

is equal to -

View answer and explanation

Correct option: A

\ell^{r+1}(\mathrm{x})+\mathrm{c}

Option A (

\ell^{r+1}(\mathrm{x})+\mathrm{c}

) is correct because the integral simplifies to the (r+1)-th iterated logarithm of x plus a constant, through a series of substitutions.

533 Easy

533 of 995

Application +4 | -1

If the anti-derivative of

\int \frac{\sin ^4 x}{x} d x

f(x)

, then

\int \frac{\sin ^4\{(p+q) x\}}{x} d x

in terms of

f(x)

View answer and explanation

Correct option: A

f\{(p+q) x\}

Option A is correct because by substituting

(p+q)x

into the antiderivative

f(x)

, we obtain

f((p+q)x)

534 Easy

534 of 995

Application +4 | -1

\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

(x>0)

is equal to

View answer and explanation

Correct option: C

e^{\tan ^{-1} x} \cdot\left(\sec ^{-1}\left(\sqrt{1+x^2}\right)\right)^2+C

Option C (

e^{\tan ^{-1} x} \cdot\left(\sec ^{-1}\left(\sqrt{1+x^2}\right)\right)^2+C

) is correct because after simplification and substitution, the integral can be solved to obtain this result.

535 Easy

535 of 995

Application +4 | -1

\int \frac{1}{(2 x-7)\sqrt{x^2-7 x+12}} d x

is equal to

View answer and explanation

Correct option: B

\sec ^{-1}(2 x-7)+C

Option B (

\sec ^{-1}(2 x-7)+C

) is correct because the integral can be simplified to a standard inverse secant form by completing the square.

536 Easy

536 of 995

Analyzing +4 | -1

The value of

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

, is

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because after algebraic manipulation and substitution, the integral simplifies to this inverse tangent form.

537 Easy

537 of 995

Application +4 | -1

\int \,\,\frac{{{a^{x/2}}}}{{\sqrt {{a^{ - x}} - \,\,{a^x}} }}\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{{\log \,\,a}}\,\,{\sin ^{ - 1}}\,\,\left( {{a^x}} \right)\,\, + \,\,c

Option A (

\frac{1}{{\log \,a}}\,{\sin ^{ - 1}}\,\left( {{a^x}} \right)\, + \,c

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

538 Easy

538 of 995

Application +4 | -1

\int \frac{{\left( {{e^x} - \,\,{e^{ - x}}} \right)\,dx}}{{\left( {{e^x} + \,\,{e^{ - x}}} \right)\log \,\,\left( {\cosh \,\,x} \right)}}

is equal to

View answer and explanation

Correct option: D

\log \,\,\left[ {\log \,\,\left( {\cosh \,\,x} \right)} \right] + c

Option D,

\log \,\left[ {\log \,\left( {\cosh \,x} \right)} \right] + c

, is correct because it results from a u-substitution where

u = \log(\cosh x)

539 Medium

539 of 995

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.

540 Easy

540 of 995

Application +4 | -1

\int\left\{\sin \left(\log _e x\right)+\cos \left(\log _e x\right)\right\} d x

is equal to

View answer and explanation

Correct option: B

x \sin \left(\log _e x\right)+C

Option B (

x \sin (\log _e x)+C

) is correct because the integral of the given expression simplifies to this form using substitution and a standard integral formula.

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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