$\smallint \,\,\cos \,\,\left[ {2\,\,{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} } \right]\,\,dx$ is equal to

$\frac{1}{2}\,\,\sin \,\,\left[ {2\,\,{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} } \right]\,\, + \,\,c$

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

$\smallint \,\,\frac{{1\,\, + \,\,x\,\, + \,\,\sqrt {x\,\, + \,\,{x^2}} }}{{\sqrt x \,\, + \,\,\sqrt {1\,\, + \,\,x} }}\,\,dx$ is equal to

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

$\sqrt {1\,\, + \,\,x} + \,\,c$

$2\,\,{\left( {1\,\, + \,\,x} \right)^{3/2}} + \,\,c$

If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to

$\frac{1}{2}\left[x^2\{F(x)\}^2-\int\{F(x)\}^2 d x\right]$

$\frac{1}{2}\left[x^2 F(x)-\frac{1}{2} \int\{F(x)\}^2 d x\right]$

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

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

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

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

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

$3\,{\tan ^{ - 1}}\,\,\sqrt {4x + 3} \,\, + \,\,c$

$4\,{\tan ^{ - 1}}\,\,\sqrt {4x + 3} \,\, + \,\,c$

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

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

$\log \,\,\left| {\,\frac{{\sin \,\,x}}{{x\,\, + \,\,\cos \,\,x}}\,} \right| + c$

$\log \,\,\left| {\,\frac{{2\,\,\sin \,\,x}}{{x\,\, + \,\,\cos \,\,x}}\,} \right| + c$

The value of $\int \frac{1+\sin x}{1-\sin x} d x$ is

$2 \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)+C$

$2 \tan ^2\left(\frac{x}{2}+\frac{\pi}{4}\right)-x+C$

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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 541-570 of 746 questions

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541 Easy

541 of 746

Application +4 | -1

\int \frac{1}{4 \cos ^3 x-3 \cos x} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{3} \ln |\sec 3 x+\tan 3 x|+c

Option B is correct because the integral simplifies to

\frac{1}{3} \ln |\sec 3 x+\tan 3 x|+c

after using the triple angle formula and integrating.

542 Easy

542 of 746

Application +4 | -1

\int \,\,\cos \,\,\left[ {2\,\,{{\cot }^{ - 1}}\,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} } \right]\,\,dx

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the expression inside the cosine simplifies to -x, and the integral of -x is -x²/2 + c.

543 Easy

543 of 746

Analyzing +4 | -1

\int \nolimits_0^2 \,\,\frac{{2x\,\, - \,\,2}}{{2x\,\, - \,\,{x^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because substituting

t = 2x - x^2

makes the limits of integration the same, resulting in a zero integral.

544 Easy

544 of 746

Application +4 | -1

\int \frac{1+\tan x}{x+\log \sec x} d x=

View answer and explanation

Correct option: A

\log (x+\log \sec x)+c

Option A is correct because the integral simplifies to

\log(x + \log \sec x) + c

by recognizing the numerator as the derivative of the denominator.

545 Easy

545 of 746

Application +4 | -1

f(x)=\frac{\log x}{x^3}

, then its antiderivative

F(x)

given by

View answer and explanation

Correct option: A

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

Option A (

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

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

546 Easy

546 of 746

Understanding +4 | -1

n

is an odd positive integer, then

\int\left|x^n\right| d x

is equal to

View answer and explanation

Correct option: C

\frac{\left|x^n\right| x}{n+1}+C

Option C is correct because, for odd n, the integral of |x^n| dx results in an expression that involves |x|^n * x / (n+1) + C, accounting for both positive and negative values of x.

547 Easy

547 of 746

Application +4 | -1

\int \sec ^{2 / 3} x \operatorname{cosec}^{4 / 3} x d x=

View answer and explanation

Correct option: B

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

Option B is correct because the integral can be simplified using trigonometric identities and solved using substitution, resulting in

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

548 Easy

548 of 746

Application +4 | -1

\int \frac{1}{1+3 \sin ^2 x} d x

is equal to

View answer and explanation

Correct option: B

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

Option B,

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

, is correct because dividing by

\cos^2 x

and using a u-substitution leads to this result.

549 Easy

549 of 746

Analyzing +4 | -1

\int \sqrt{x-3}\left\{\sin ^{-1}(\ln x)+\cos ^{-1}(\ln x)\right\} d x

is equal to

View answer and explanation

Correct option: C

Does not exist

The correct answer is D (Does not exist) because the domain of

\sin^{-1}(\ln x) + \cos^{-1}(\ln x)

is restricted to

-1 \leq \ln x \leq 1

, which is not always true.

550 Easy

550 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral simplifies to

\int \sqrt{1+x} dx

, which integrates to the given expression.

551 Easy

551 of 746

Application +4 | -1

\int \,\,\frac{{1\,\, + \,\,\tan \,\,x}}{{{e^{ - x}}\,\,\cos \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the integral simplifies to this expression after applying integration by parts.

552 Easy

552 of 746

Application +4 | -1

\int f(x) d x=F(x)

, then

\int x^3 f\left(x^2\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\left[x^2 F\left(x^2\right)-\int F\left(x^2\right) d\left(x^2\right)\right]

Option B is correct. The integral is solved using substitution and integration by parts, resulting in the given expression.

553 Easy

553 of 746

Application +4 | -1

\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right)} \,{e^{({e^x} + {e^{ - x}})}}dx = g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c,

where

c

is a constant of integration, then

g\left( 0 \right)

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because after solving the integral and substituting x=0 into the resulting function g(x), we obtain the value 2.

554 Easy

554 of 746

Application +4 | -1

|\ln x| \mathrm{d} x

equals

(0 < x < 1)

View answer and explanation

Correct option: A

x+x|\ln x|+c

Option B (

x|\ln x|-x+c

) is correct because integrating |ln x| with the given condition 0 < x < 1 results in the expression x|ln x| - x + c.

555 Easy

555 of 746

Application +4 | -1

\int \tan ^{-1} \sqrt{\frac{1-x}{1+x}} d x=

View answer and explanation

Correct option: A

\frac{1}{2}\left(x \cos ^{-1} x-\sqrt{1-x^2}\right)+C

Option A (

\frac{1}{2}\left(x \cos ^{-1} x-\sqrt{1-x^2}\right)+C

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

556 Easy

556 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

\frac{x e^x}{x+4}+c

Option A (

\frac{x e^x}{x+4}+c

) is correct because the integral is solved using the form

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

557 Easy

557 of 746

Application +4 | -1

b>a

, and

I=\int_a^b \sqrt{\frac{x-a}{b-x}} d x

, then I equals

View answer and explanation

Correct option: A

\frac{\pi}{2}(b-a)

Option A (

\frac{\pi}{2}(b-a)

) is correct because the integral evaluates to this value after a suitable substitution and trigonometric simplification.

558 Easy

558 of 746

Application +4 | -1

\int \frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x} d x

is equal to

View answer and explanation

Correct option: C

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

Option C (-

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

) is correct because it is the simplified result after applying trigonometric identities and integration.

559 Easy

559 of 746

Application +4 | -1

\begin{aligned} & \int\left(x^{2010}+x^{804}+x^{402}\right)\left(2 x^{1608}+5 x^{402}+10\right)^{\frac{1}{402}} d x \\ & =\frac{1}{10 a}\left(2 x^{2010}+5 x^{804}+10 x^{402}\right)^{\frac{a}{402}}, \text { then } a-400 \text { is } \end{aligned}

View answer and explanation

Correct option: A

Option A (3) is correct because it is the value of a - 400, derived from the given integral by substitution and simplification.

560 Easy

560 of 746

Application +4 | -1

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

View answer and explanation

Correct option: D

e^x \tan x+c

Option D (

e^x \tan x+c

) is correct because the integral simplifies to this expression by recognizing the form of

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

561 Easy

561 of 746

Application +4 | -1

\int \frac{\sin x \cos x}{a \cos ^2 x+b \sin ^2 x} d x=

View answer and explanation

Correct option: A

\frac{1}{2(b-a)} \log \left(a \cos ^2 x+b \sin ^2 x\right)+c

Option A is correct because the integral can be solved using substitution.

562 Easy

562 of 746

Application +4 | -1

\int e^x \sin x d x=\frac{1}{2} e^x \cdot a+c

, then

a=

View answer and explanation

Correct option: A

\sin x-\cos x

Option A (

\sin x-\cos x

) is correct because it is the result of solving the integral

\int e^x \sin x dx

using integration by parts twice and comparing with the given form.

563 Easy

563 of 746

Application +4 | -1

\int x^3 \log x d x=

View answer and explanation

Correct option: B

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

Option B is correct because the integral of x³ log(x) dx is found using integration by parts, resulting in 1/16[4x⁴ log(x) - x⁴] + c.

564 Easy

564 of 746

Application +4 | -1

\int \sin 2 x \log _e \cos x d x

is equal to

View answer and explanation

Correct option: C

\left(\frac{1}{2}-\log _e \cos x\right) \cos ^2 x+C

Option C,

\left(\frac{1}{2}-\log _e \cos x\right) \cos ^2 x+C

, is correct because it results from integrating

\sin 2x \log_e \cos x

using integration by parts and substitution.

565 Easy

565 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\sqrt {{x^2} - \,\,1} - \,\,{\sin ^{ - 1}}\,\,x + c

Option B (

\sqrt {{x^2} - \,1} - \,{\sin ^{ - 1}}\,x + c

) is correct because it is the result of simplifying the integrand and using appropriate integration techniques.

566 Easy

566 of 746

Application +4 | -1

\int \frac{{dx}}{{\left( {x + 1} \right)\,\,\sqrt {4x + 3} }}

is equal to

View answer and explanation

Correct option: C

2\,{\tan ^{ - 1}}\,\,\sqrt {4x + 3} \,\, + \,\,c

Option C (

2\,{\tan ^{ - 1}}\,\sqrt {4x\, + \,3} \, + \,c

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

567 Easy

567 of 746

Understanding +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D,

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

, is correct because the integral simplifies to this form after algebraic manipulation and recognizing the derivative in the denominator.

568 Easy

568 of 746

Application +4 | -1

The value of

\int \frac{1+\sin x}{1-\sin x} d x

View answer and explanation

Correct option: C

2 \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)-x+C

Option C (

2 \tan \left(\frac{x}{2}+\frac{\pi}{4}\right)-x+C

) is correct because it is the result of integrating the given expression and simplifying using trigonometric identities.

569 Easy

569 of 746

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\sin 2 x+C

Option A (

\sin 2 x+C

) is correct because the integral simplifies to

2 \int \cos 2x dx

, which equals

\sin 2x + C

570 Easy

570 of 746

Application +4 | -1

\int e^x \tan ^2\left(e^x\right) d x=

View answer and explanation

Correct option: D

\tan \left(e^x\right)-e^x+c

Option D (

\tan \left(e^x\right)-e^x+c

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

Revision notes and key takeaways

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