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

${\sin ^{ - 1}}\,\,\left( {2x\,\, + \,\,5} \right)\,\, + \,\,c$

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

${\sin ^{ - 1}}\,\,\left( {5\,\, - \,\,2x} \right)\,\, + \,\,c$

$\smallint \,\frac{{{x^2}}}{{1\,\, + \,\,{x^6}}}\,dx$ is equal to

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

$\smallint \,(1 - \cos \,\,x)\;\,{\rm{cose}}{{\rm{c}}^2}\,\,dx$ is equal to

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

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

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

If $I=\int(\sqrt{\tan x}+\sqrt{\cot x}) d x$, then I equals:

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

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

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

$\int \frac{x^3-x^2+x-1}{x^5+1} d x$ is

$\frac{1}{5} \ln \left(\frac{1+x^5}{\left(1+x^3\right)^5}\right)+c$

$\frac{1}{5} \ln \left(\frac{1+x^5}{1+x^3}\right)+c$

$\frac{1}{5} \ln \left(\frac{1+x^5}{1+x}\right)+c$

$\smallint \,{e^{\sin {\rm{\theta }}}}\,[\log \,\,\sin \,\,{\rm{\theta }} + \,\,{\rm{cose}}{{\rm{c}}^2}\,{\rm{\theta }}]\,\,\cos \,\,{\rm{\theta }}\,d{\rm{\theta }}$ is equal to

${e^{\sin {\rm{\theta }}}}\,\,\left[ {\log \,\,\sin \,\,{\rm{\theta }}\,\, - \,\,{\rm{cosec}}\,\,{\rm{\theta }}} \right]\,\, + \,\,c$

$\smallint \,\,{e^{\sin {\rm{\theta }}}}\,\,[\log \,\,\sin \,\,{\rm{\theta }}\,\, + \,\,{\rm{cose}}{{\rm{c}}^2}\,\,{\rm{\theta }}]\,\, + \,\,c$

${e^{\sin {\rm{\theta }}}}\,\,\left[ {\log \,\,\sin \,\,{\rm{\theta }}\,\, + \,\,{\rm{cosec}}\,\,{\rm{\theta }}} \right]\,\, + \,\,c$

${e^{\sin \theta }}\,\,\left[ {\log \,\,\sin \,\,\theta \,\, - \,\,{\rm{cose}}{{\rm{c}}^2}\,\theta } \right]\,\, + \,\,c$

$\smallint \,\,x{e^{{x^2}}}\,\,dx$ is equal to

$\frac{{{e^x}}}{2}\,\, + \,\,c$

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Easy

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Practice format and chapter quiz

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 601-630 of 746 questions

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601 Easy

601 of 746

Application +4 | -1

The function

f

whose graph passes through

(0,7 / 3)

and whose derivative is

x \sqrt{1-x^2}

is given by

View answer and explanation

Correct option: C

f(x)=-(1 / 3)\left[\left(1-x^2\right)^{3 / 2}-8\right]

Option C (

f(x)=-(1 / 3)\left[\left(1-x^2\right)^{3 / 2}-8\right]

) is correct because integrating

x \sqrt{1-x^2}

and applying the initial condition

f(0) = 7/3

yields the correct function.

602 Easy

602 of 746

Analyzing +4 | -1

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

then

I

is equal to

View answer and explanation

Correct option: A

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

The correct answer is

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

because it fits the form of the integral

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

603 Easy

603 of 746

Application +4 | -1

I=\int_0^1 x \sqrt{\frac{1-x}{1+x}} d x

, the I equals

View answer and explanation

Correct option: B

1-\frac{\pi}{4}

Option C is correct because splitting the integral and using substitution simplifies it to a manageable form, resulting in 2(1 - 1/e).

604 Easy

604 of 746

Application +4 | -1

\int \,\,\frac{{dx}}{{\sqrt {\left( {1 - x} \right)\,\,\left( {x - 2} \right)} }}

is equal to

View answer and explanation

Correct option: A

{\sin ^{ - 1}}\,\,\left( {2x\,\, - \,\,3} \right)\,\, + \,\,c

Option A (

{\sin ^{ - 1}}\,\,\left( {2x\, - \,3} \right)\,\,+ \,c

) is correct because it is the result of completing the square and applying the standard integral formula for arcsine.

605 Easy

605 of 746

Application +4 | -1

\int \,\,{e^{3\log x}}\,\,{\left( {{x^4} + \,\,1} \right)^{ - 1}}dx

is equal to

View answer and explanation

Correct option: B

\frac{1}{4}\,\,\log \,\,\left( {{x^4}\, + \,\,1} \right) + C

Option B (

\frac{1}{4}\log \left( {{x^4} + 1} \right) + C

) is correct because it is the result of evaluating the indefinite integral by simplifying the expression and using a simple substitution.

606 Easy

606 of 746

Application +4 | -1

\int \,\frac{{{x^2}}}{{1\,\, + \,\,{x^6}}}\,dx

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because by substituting

t = x^3

, the integral simplifies to a standard arctangent integral.

607 Easy

607 of 746

Application +4 | -1

\int \,(1 - \cos \,\,x)\;\,\mathrm{cose}{\mathrm{c}^2}\,\,dx

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the integral simplifies to

\int \frac{1}{{1 + \cos x}} dx

, which evaluates to

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

608 Easy

608 of 746

Application +4 | -1

I=\int(\sqrt{\tan x}+\sqrt{\cot x}) d x

, then I equals:

View answer and explanation

Correct option: C

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

Option C (

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

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

609 Easy

609 of 746

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 (1) is correct because, after solving the integral and comparing coefficients, the absolute difference between m and n is 1.

610 Easy

610 of 746

Application +4 | -1

\int \frac{\sin x}{\sin (x-\alpha)} d x

=A x+B \log \sin (x-\alpha)+C

, then the value of

(A, B)

, is

View answer and explanation

Correct option: B

(\cos \alpha, \sin \alpha)

Option B (

\cos \alpha, \sin \alpha

) is correct because, after simplifying the integral using trigonometric identities, the coefficients of

x

and

\log \sin(x-\alpha)

are

\cos \alpha

and

\sin \alpha

respectively.

611 Easy

611 of 746

Analyzing +4 | -1

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

View answer and explanation

Correct option: C

\frac{1}{5} \ln \left(\frac{1+x^5}{(1+x)^5}\right)+c

Option C (

\frac{1}{5} \ln \left(\frac{1+x^5}{(1+x)^5}\right)+c

) is correct because the integral can be simplified and then integrated term by term.

612 Easy

612 of 746

Analyzing +4 | -1

\int\left(x+\sqrt{1+x^2}\right)^n d x.

=\frac{1}{a(n+1)}\left\{x+\sqrt{1+x^2}\right\}^{n+1}+\frac{1}{-b(n-1)}\left\{x+\sqrt{1+x^2}\right\}^{n-1}+C

Then

(a+b)

is equal to

View answer and explanation

Correct option: C

Option C is correct because after evaluating the integral and comparing coefficients, the value of (a+b) is found to be 4.

613 Easy

613 of 746

Application +4 | -1

Let

I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x

. If

I(0)=3

, then

{I}\left(\frac{\pi}{12}\right)

is equal to :

View answer and explanation

Correct option: B

3 \sqrt{3}

Option D (

3 \sqrt{3}

) is correct because after evaluating the integral and applying the given initial condition, we find the value of

I(\frac{\pi}{12})

to be

3\sqrt{3}

614 Easy

614 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

\log \left[\frac{1+e^x}{2+e^x}\right]+c

Option B is correct because the integral is solved by substituting

e^x = t

and using partial fraction decomposition.

615 Easy

615 of 746

Analyzing +4 | -1

\int \frac{m x^{m+2 n-1}-n x^{n-1}}{x^{2 m+2 n}+2 x^{m+n}+1} d x

is equal to

View answer and explanation

Correct option: C

\frac{-x^n}{x^{m+n}+1}+C

Option C (

\frac{-x^n}{x^{m+n}+1}+C

) is correct because integrating by parts and simplifying the resulting terms leads to this expression.

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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

616 Easy

616 of 746

Analyzing +4 | -1

\int \frac{1-7 \cos ^2 x}{\sin ^7 x \cos ^2 x} d x=\frac{f(x)}{(\sin x)^7}+C

, then

f(x)

is equal to

View answer and explanation

Correct option: C

\tan x

Option C (

\tan x

) is correct because after simplifying and integrating,

f(x)

is found to be

\tan x

617 Easy

617 of 746

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{3}{4}\left(\frac{2+x}{2-x}\right)^{2 / 3}+c

Option C is correct because the integral is solved using substitution, where (2-x)/(2+x) = t³, simplifying the integral to a power rule.

618 Easy

618 of 746

Analyzing +4 | -1

\int \frac{4 x+1}{x^2+3 x+2} d x=a \log |x+1|+b \log |x+2|+C

, then

View answer and explanation

Correct option: B

a+b=4

Option B is correct because after performing partial fraction decomposition and integrating, we find that

a + b = 4

619 Easy

619 of 746

Application +4 | -1

\int \frac{1}{\cos ^{-1} x \cdot \sqrt{1-x^2}} d x=

View answer and explanation

Correct option: B

-\log \left(\cos ^{-1} x\right)+c

Option B (

-\log \left(\cos ^{-1} x\right)+c

) is correct because the integral simplifies to this form after a u-substitution.

620 Easy

620 of 746

Analyzing +4 | -1

\int \,\,\frac{{x + 2}}{{2{x^2} + \,\,6x\,\, + 5}}\,\,dx = P\,\,\int \,\,\frac{{4x + 6}}{{2{x^2} + \,\,6x\,\, + 5}}\,\,dx\,\, + \,\,\frac{1}{2}\int \,\,\frac{{dx}}{{2{x^2} + \,\,6x\,\, + 5}}

, then the value of

P

View answer and explanation

Correct option: C

\frac{1}{4}

Option C (

\frac{1}{4}

) is correct because by manipulating the integral, we can find that P equals

\frac{1}{4}

621 Easy

621 of 746

Understanding +4 | -1

If it is known that at the point

x=1

two antiderivative of

f(x)=e^x

differ by 2 . The difference of these antiderivatives at

x=100

View answer and explanation

Correct option: C

Option C (2) is correct because the difference between two antiderivatives of the same function is a constant, and this constant is given as 2 at x=1, so it remains 2 at x=100.

622 Easy

622 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A,

\frac{1}{2} \log (x-1)-\frac{1}{4} \log (x^2+1)-\frac{1}{2} \tan ^{-1} x+c

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

623 Easy

623 of 746

Application +4 | -1

\int \frac{\cos x-\sin x}{1+\sin 2 x} d x=

View answer and explanation

Correct option: A

-\frac{1}{\cos x+\sin x}+c

Option A is correct because the integral simplifies to

-\frac{1}{\cos x + \sin x} + c

after recognizing the denominator as

(\sin x + \cos x)^2

and using substitution.

624 Easy

624 of 746

Application +4 | -1

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

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 is the result of integrating the given expression using substitution and integration by parts.

625 Easy

625 of 746

Understanding +4 | -1

\int x^x\left(1+\log _e x\right) d x

is equal to

View answer and explanation

Correct option: C

x^x+C

Option C is correct because the integral can be simplified by recognizing that

x^x(1 + \log_e x)

is the derivative of

x^x

626 Easy

626 of 746

Application +4 | -1

\int \,{e^{\sin \mathrm{\theta }}}\,[\log \,\,\sin \,\,\mathrm{\theta } + \,\,\mathrm{cose}{\mathrm{c}^2}\,\mathrm{\theta }]\,\,\cos \,\,\mathrm{\theta }\,d\mathrm{\theta }

is equal to

View answer and explanation

Correct option: C

{e^{\sin \mathrm{\theta }}}\,\,\left[ {\log \,\,\sin \,\,\mathrm{\theta }\,\, - \,\,\mathrm{cosec}\,\,\mathrm{\theta }} \right]\,\, + \,\,c

The correct answer is e^{(sin θ)} [log sin θ - cosec θ] + c because integrating by parts and simplifying leads to this result.

627 Easy

627 of 746

Application +4 | -1

\int \,\,x{e^{{x^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{{e^{{x^2}}}}}{2}\,\, + \,\,c

Option B is correct because by substituting x² = t, the integral is simplified and evaluated to be e^(x²)/2 + c.

628 Easy

628 of 746

Application +4 | -1

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

equals:

View answer and explanation

Correct option: B

\frac{1}{3} \tan ^{-1} x-\frac{1}{6} \tan ^{-1} \frac{x}{2}+c

Option B is correct because the integral is solved using partial fraction decomposition, separating the integrand into simpler fractions that can be easily integrated.

629 Easy

629 of 746

Analyzing +4 | -1

\int \frac{1}{a^2 \sin ^2 x+b^2 \cos ^2 x} d x=\frac{1}{12} \tan ^{-1}(3 \tan x)+C,

then the maximum value of

a \sin x+b \cos x

, is

View answer and explanation

Correct option: B

\sqrt{40}

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

ab = 12

and

\frac{a}{b} = 3

, leading to the maximum value of

a \sin x + b \cos x

being

\sqrt{40}

630 Easy

630 of 746

Application +4 | -1

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

equals

View answer and explanation

Correct option: A

\sec ^{-1}\left(x^x\right)+C

Option A is correct because the integral simplifies to sec^(-1)(x^x) + C after recognizing the derivative of x^x.

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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