$\smallint \,\,\frac{{dx}}{{{a^2}\,\,{{\sin }^2}\,\,x\,\, + \,\,{b^2}\,\,{{\cos }^2}\,\,x}}$ is equal to

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

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

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

$\frac{{ - 1}}{{\sin \,\,x\,\, + \,\,\cos \,\,x}}\,\, + \,\,C$

$\log {\left( {\sin \,\,x\,\, + \,\,\cos \,\,x} \right)^2}\,\, + \,\,C$

$\smallint \,\,\sqrt x e\,{\,^{\sqrt x }}dx$ is equal to

$2\,\sqrt x - {e^{\sqrt x }} - \,\,4\,\sqrt {x{e^{\sqrt x }}} + \,\,c$

$\left( {1\,\, - \,\,4\,\,\sqrt x } \right){e^{\sqrt x }} + \,\,c$

If $\smallint f\,\,\left( x \right)\,\,dx = {\rm{g}}\,\,\left( x \right)$, then $f\,\,\left( x \right)\,\,{\rm{g}}\left( x \right)$ is equal to

$\frac{1}{2}\,\,{{\rm{g}}^2}\left( x \right)$

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

$\frac{1}{2}\,\,{\left[ {{\rm{g'}}\,\,\left( x \right)} \right]^2}$

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

The value of $\smallint \,\,\frac{{{{\left( {x\,\, - \,\,{x^3}} \right)}^{1/3}}}}{{{x^4}}}\,\,dx$ is

$\frac{1}{8}\,\,{\left( {1\,\, - \,\,\frac{1}{{{x^2}}}} \right)^{4/3}} + \,\,1$

$\smallint {x^2}\,\,{\left( {ax + \,\,b} \right)^{ - 2}}\;\,dx$ is equal to

$\frac{2}{{{a^2}}}\,\,\left( {x\,\, - \,\,\frac{b}{a}\,\,\log \,\,\left( {ax\,\, + \,\,b} \right)} \right)\,\, + \,\,c\;$

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

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

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

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

$\frac{1}{4}\,\,\log \,\,\left( {1 + \sqrt x \,} \right) + c$

$3\,\,\log \,\,\left( {1 + \sqrt x \,} \right) + c$

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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 421-450 of 746 questions

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421 Easy

421 of 746

Application +4 | -1

I=\int \frac{1}{\left(a^2-b^2 x^2\right)^{3 / 2}} d x

is equal to

View answer and explanation

Correct option: B

\frac{x}{a^2 \sqrt{a^2-b^2 x^2}}+C

Option B is correct because the integral simplifies to x / (a² * sqrt(a² - b² * x²)) + C after trigonometric substitution.

422 Easy

422 of 746

Application +4 | -1

\int \,\,\frac{{dx}}{{{a^2}\,\,{{\sin }^2}\,\,x\,\, + \,\,{b^2}\,\,{{\cos }^2}\,\,x}}

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the integral can be solved by dividing by cos²(x) and using a suitable substitution.

423 Easy

423 of 746

Application +4 | -1

\int \frac{d x}{5+4 \cos x}=

View answer and explanation

Correct option: C

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

Option C is correct because the integral is solved using the tangent half-angle substitution, resulting in

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

424 Easy

424 of 746

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 it represents the result of the given integral after applying integration by parts and simplification.

425 Easy

425 of 746

Application +4 | -1

\int \frac{{\cos \,\,2x}}{{{{\left( {\sin \,\,x\,\, + \,\,\cos \,\,x} \right)}^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\log \left( {\sin \,x\,\, + \,\,\cos \,\,x} \right)\,\, + \,\,C

Option B is correct because the integral simplifies to a logarithmic function after simplification.

426 Easy

426 of 746

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C (

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

) is the correct result obtained by using substitution and integrating the resulting trigonometric function.

427 Easy

427 of 746

Application +4 | -1

\int \frac{d x}{\left(x-x^2\right)}=

View answer and explanation

Correct option: A

\log x-\log (1-x)+c

Option C (

\log x-\log (1-x)+c

) is correct because the integral is solved by partial fraction decomposition.

428 Easy

428 of 746

Application +4 | -1

\int 2 x \cos ^3 x^2 \sin x^2 d x=

View answer and explanation

Correct option: A

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

Option A (

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

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

429 Easy

429 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the result of the integration after using a suitable substitution.

430 Easy

430 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A is correct because the integral simplifies to

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

after substitution and integration.

431 Easy

431 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\left( {2x\,\, - \,\,4\,\sqrt x + 4} \right){e^{\sqrt x }} + \,\,c

Option B (

\left( {2x\, - \,4\,\sqrt x + 4} \right){e^{\sqrt x }} + \,c

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

432 Easy

432 of 746

Application +4 | -1

\int f\,\,\left( x \right)\,\,dx = \mathrm{g}\,\,\left( x \right)

, then

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

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\,\,{\mathrm{g}^2}\left( x \right)

Option B (

\frac{1}{2}\,{\mathrm{g}^2}\left( x \right)

) is correct because after applying integration by parts and simplifying, the result matches this option.

433 Easy

433 of 746

Application +4 | -1

\int \frac{1}{x^{1 / 5}\left(1+x^{4 / 5}\right)^{1 / 2}} d x

, is

View answer and explanation

Correct option: B

\frac{5}{2} \sqrt{1+x^{4 / 5}}+C

Option B (

\frac{{5}}{{2}} \sqrt{{1+x^{4 / 5}}}+C

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

434 Easy

434 of 746

Application +4 | -1

The value of

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

View answer and explanation

Correct option: B

- \frac{3}{8}\,\,{\left( {\frac{1}{{{x^2}}}\,\, - \,\,1} \right)^{4/3}} + \,\,C

Option B (

\ - \frac{3}{8}\,{\left( {\frac{1}{{{x^2}}}\,\, - \,\,1} \right)^{4/3}} + \,\,C

) is correct because after simplifying the integrand and using substitution, the result matches this option.

435 Easy

435 of 746

Analyzing +4 | -1

\int {x^2}\,\,{\left( {ax + \,\,b} \right)^{ - 2}}\;\,dx

is equal to

View answer and explanation

Correct option: B

\frac{2}{{{a^2}}}\,\,\left( {x\,\, - \,\,\frac{b}{a}\,\,\log \,\,\left( {ax\,\, + \,\,b} \right)} \right)\,\, - \,\,\frac{{{x^2}}}{{a\left( {ax\,\, + \,\,b} \right)}}\,\, + \,\,c

The correct answer is D (

\frac{2}{{{a^2}}}\,\\,\,\,\left( {x\,\, + \,\,\frac{b}{a}\,\\,\,\log \,\,\,\left( {ax\,\, + \,\,b} \right)} \right)\,\, - \,\,\,\frac{{{{{x^2}}}}}{{{{a\left( {ax\,\, + \,\,b} \right)}}}}\,\, + \,\,c

) because it results from correct substitution and integration by parts.

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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

436 Easy

436 of 746

Application +4 | -1

The integral

\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x

equals (for some arbitrary constant

K

)

View answer and explanation

Correct option: C

-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K

Option C (

-\frac{1}{(\sec x+\tan x)^{11 / 2}}\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\}+K

) is correct because substituting

t = \sec x + \tan x

and simplifying leads to the given result.

437 Easy

437 of 746

Application +4 | -1

\int \tan (3 x-5) \sec (3 x-5) d x=

View answer and explanation

Correct option: B

\frac{1}{3} \sec (3 x-5)+c

Option B is correct because substituting

t = 3x - 5

transforms the integral to

\frac{1}{3} \int \tan t \sec t dt

, which equals

\frac{1}{3} \sec t + c = \frac{1}{3} \sec(3x - 5) + c

438 Easy

438 of 746

Application +4 | -1

\int\{\sqrt{\tan x}+\sqrt{\cot x}\} d x

is equal to

View answer and explanation

Correct option: B

\sqrt{2} \sin ^{-1}(\sin x-\cos x)+C

Option B,

\sqrt{2} \sin ^{-1}(\sin x-\cos x)+C

, is correct because manipulating the integrand and using a trigonometric identity leads to this result.

439 Easy

439 of 746

Analyzing +4 | -1

\int \,\,{\left\{ {1\,\, + \,\,2\,\,\tan \,\,x\,\,\left( {\tan \,\,x\,\, + \,\,\sec \,\,x} \right)} \right\}^{1/2}}dx

is equal to

View answer and explanation

Correct option: C

\log \,\,\sec \,\,x\,\left( {\sec \,\,x\,\, + \,\,\tan \,\,x\,\, + \,\,C} \right)

Option C is correct because the integral simplifies to \

\log \sec x + \log (\sec x + \tan x)

, which can be combined to give \

\log \sec x(\sec x + \tan x)

440 Easy

440 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

-\sqrt{2} \tan ^{-1} \frac{x}{\sqrt{2}}+\sqrt{3} \tan ^{-1} \frac{x}{\sqrt{3}}+c

Option B,

-\sqrt{2} \tan ^{-1} \frac{x}{\sqrt{2}}+\sqrt{3} \tan ^{-1} \frac{x}{\sqrt{3}}+c

, is the correct answer because it results from partial fraction decomposition and integration of the given expression.

441 Easy

441 of 746

Understanding +4 | -1

\int \,\,{e^x}\,\,\left[ {f\left( x \right) + f'\left( x \right)} \right]\,\,dx

is equal to

View answer and explanation

Correct option: A

{e^x}f\,\,\left( x \right) + c

Option A (

{e^x}f\,\left( x \right) + c

) is correct because the integral of

{e^x}[f(x) + f'(x)] dx

results in

{e^x}f(x) + c

using integration by parts.

442 Easy

442 of 746

Analyzing +4 | -1

\int \frac{1}{\sin ^6 x+\cos ^6 x} d x

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because after simplifying and integrating, the result matches this option.

443 Easy

443 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B is correct because the integral of arccos(1/x) dx is equivalent to the integral of arcsec(x) dx, which after integration by parts results in x arcsec(x) - arccosh(x) + C.

444 Easy

444 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A,

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

, is correct because it is the result of integrating the given function using substitution and partial fractions.

445 Easy

445 of 746

Application +4 | -1

\int \left( {\sin \,\,2x - \cos \,\,\,2x} \right)dx = \frac{1}{{\sqrt 2 }}\,\,\sin \,\,\left( {2x - a} \right) + b

, then

View answer and explanation

Correct option: B

a = - \frac{{5\pi }}{4},\,\;b \in R

Option B (

a = - \frac{{5\pi }}{4},\;b \in R

) is correct because it is derived by integrating the given trigonometric expression and comparing it with the target form.

446 Easy

446 of 746

Application +4 | -1

The area under the curve

y=x^2

and the line

x=3

and

\mathrm{x}

axis is_____ sq. units.

View answer and explanation

Correct option: A

Option D (1) is correct because setting up the integral and solving for 'a' gives a value of 1.

447 Easy

447 of 746

Analyzing +4 | -1

\int \sqrt{\frac{x}{a^3-x^3}} d x=m \sin ^{-1}\left(\frac{x}{a}\right)^n+C

, then

View answer and explanation

Correct option: C

m=1 / n

Option C is correct because after evaluating the integral, we find that

m = \frac{2}{3}

and

n = \frac{3}{2}

, which means

m = \frac{1}{n}

448 Easy

448 of 746

Application +4 | -1

\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x

equals to (for some arbitrary constant

K

)

View answer and explanation

Correct option: C

\frac{-1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K

Option C (

\frac{{-1}}{{(\sec x+\tan x)^{{11 / 2}}}}\left{\frac{{1}}{{11}}+\frac{{1}}{{7}}(\sec x+\tan x)^2\right\}+K

) is correct because it represents the result of the given integral after applying substitution and simplification.

449 Easy

449 of 746

Application +4 | -1

\int x^n \log x d x=

View answer and explanation

Correct option: D

\frac{x^{n+1}}{n+1}\left\{\log x-\frac{1}{n+1}\right\}+c

Option D (

\frac{x^{n+1}}{n+1}\left{\log x-\frac{1}{n+1}\right}+c

) is correct because it is the result of integrating

x^n \log x

using integration by parts.

450 Easy

450 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

2\,\,\log \,\,\left( {1 + \sqrt x \,} \right)\, + c

Option B (

2\,\log \,\left( {1 + \sqrt x \} \right)\, + c

) is correct because after substitution, the integral simplifies and evaluates to this expression.

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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