The value of $\smallint {e^x}\left( {{x^5} + \,\,5{x^4} + \,\,1} \right)dx$ is

${e^x}\,\,.\,\,{x^5}\,\, + \,\,c$

${e^{x + 1}}\,\,.\,\,{x^5}\,\, + \,\,c$

$5{x^4}\,\,.\,\,{e^x}\,\, + \,\,c$

$\smallint \,\,\left| {\,x\,} \right|\,\,dx$ is equal to

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

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

$\smallint \,\,\frac{{\sqrt {\tan \,\,x} }}{{\sin \,\,x\,\,\cos \,\,x}}\,\,dx$ is equal to

$2\,\sqrt {\cot \,\,x} \,\, + \,\,C$

$\frac{{\sqrt {\tan \,\,x} }}{2}\,\, + \,\,C$

Let $\smallint \,\,{e^x}\,\,\left( {f\,\,\left( x \right) - f'\left( x \right)} \right)\,\,dx = \phi \,\,\left( x \right)$ Then, $\smallint \,\,{e^x}\,f\left( x \right)\,\,dx$ is equal to

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

$\smallint \left( x \right)\,\, - \,\,{e^x}f\,\,\left( x \right)$

$\smallint \,\,\frac{{1\,\, + \,\,x}}{{1\,\, + \,\,\sqrt[3]{x}}}\,\,dx$ is equal to

$\frac{3}{5}\,\,{x^{5/3}}\,\, - \,\,\frac{3}{4}\,\,{x^{4/3}}\,\, + \,\,C$

\[\smallint \frac{{\sec \,\,x}}{{\sec \,\,x\,\, + \,\,\tan x}}\,\,\;dx\] is equal to

$\sec \,\,x\,\, + \,\,\tan \,\,x + c$

$\log \,\,\sin \,\,x + \log \,\,\cos \,\,x + c$

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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 691-720 of 746 questions

691 Easy

691 of 746

Analyzing +4 | -1

\int {\frac{{({x^2} + 1){e^x}}}{{{{(x + 1)}^2}}}dx = f(x){e^x} + C,}

where C is a constant then

\frac{{{d^3}f}}{{d{x^3}}}

x = 1

is equal to:

View answer and explanation

Correct option: B

\frac{3}{4}

Option B (

\frac{3}{4}

) is correct because after finding f(x) and its derivatives, substituting x=1 into the third derivative gives

\frac{3}{4}

692 Easy

692 of 746

Application +4 | -1

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

equals

View answer and explanation

Correct option: A

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

Option A is correct because the integral is solved using substitution, where x^2-1 = t², simplifying the integral to a rational function.

693 Easy

693 of 746

Understanding +4 | -1

\int \frac{\operatorname{cosec} x}{\log \tan \frac{x}{2}} d x=

View answer and explanation

Correct option: A

\log \left(\log \tan \frac{x}{2}\right)+c

Option A (

\log \left(\log \tan \frac{x}{2}\right)+c

) is correct because the derivative of the inner function

\log \tan \frac{x}{2}

is a constant multiple of

\operatorname{cosec} x

694 Easy

694 of 746

Application +4 | -1

\int \frac{2^{1 / x}}{x^2} d x=a\, 2^{1 / x}+C

, then

a=

View answer and explanation

Correct option: A

-\log _2 e

Option A (

- \log _2 e

) is correct because, after simplifying the integral, the coefficient of

2^{1/x}

-\log_2 e

695 Easy

695 of 746

Application +4 | -1

\int \cos \,\,({\log _e}\,\,x)\,dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{2}\,\,x\,[\cos \,\,({\log _e}\,\,x)\,\, + \,\,\sin \,\,({\log _e}\,\,x)]

Option A (

\frac{1}{2}\,x[\cos \,({\log _e}\,x)\, + \,\sin \,({\log _e}\,x)]

) is correct because it is the result of applying integration by parts twice.

696 Easy

696 of 746

Application +4 | -1

The equation of a curve passing through the point

(0,1)

be given by

y=\int x^2 \cdot e^{x^3} d x

. If the equation of the curve be written in the form

x=f(y)

, then

f(y)=

View answer and explanation

Correct option: B

\sqrt[3]{\log _e(3 y-2)}

Option B is correct because it involves solving the integral, applying the initial condition, and then finding the inverse function.

697 Easy

697 of 746

Application +4 | -1

\int g(x) d x=g(x)

, then the value of the integral

\int f(x) g(x)\left\{f(x)+2 f^{\prime}(x)\right\} d x

, is

View answer and explanation

Correct option: B

\{f(x)\}^2 g(x)+C

The correct answer is B (

\{f(x)\}^2 g(x)+C

) because it results from integrating by parts and using the given condition

\int g(x) dx = g(x)

698 Easy

698 of 746

Application +4 | -1

If the integral

\int \frac{5 \tan x}{\tan x-2} d x=x+

a \log |\sin x-2 \cos x|+K

, then

a

is equal to

View answer and explanation

Correct option: C

Option C is correct because the integral can be solved by rewriting the integrand and using logarithmic integration.

699 Easy

699 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

3\left\{\log \left(\frac{x^{1 / 3}}{1+x^{1 / 3}}\right)+\frac{1}{1+\sqrt[3]{x}}\right\}+C

Option A,

3\left{\log \left(\frac{x^{1 / 3}}{1+x^{1 / 3}}\right)+\frac{1}{1+\sqrt[3]{x}}\right\}+C

, is correct because it is the result of integrating the given expression after a suitable substitution and partial fraction decomposition.

700 Easy

700 of 746

Application +4 | -1

The value of

\int {e^x}\left( {{x^5} + \,\,5{x^4} + \,\,1} \right)dx

View answer and explanation

Correct option: B

{e^x}\,\,.\,\,{x^5}\,\, + \,\,{e^x} + \,\,c

Option B (

{e^x}\,.\,{x^5}\, + \,{e^x} + \,c

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

701 Easy

701 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

\sin ^{-1}(x-1)+c

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

\sin^{-1}(x-1)+c

702 Easy

702 of 746

Application +4 | -1

\int \,\,\frac{{dx}}{{{x^2} + 2x + 2}} = f\,\left( x \right) + c

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because completing the square and using the standard integral formula for arctangent yields this result.

703 Easy

703 of 746

Application +4 | -1

\int {\frac{{dx}}{{x\ln x\ln \left( {\ln x} \right)}}}

is equal to

View answer and explanation

Correct option: A

\ln \left| {\left( {\ln \left( {\ln x} \right)} \right)} \right| + c

Option A (

\ln \left| {\left( {\ln \left( {\ln x} \right)} \right)} \right| + c

) is correct because after two successive substitutions, the integral simplifies to this logarithmic expression.

704 Easy

704 of 746

Understanding +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the integral of

|x|

\frac{1}{2}x|x| + c

705 Easy

705 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

\frac{e^x}{2 x+3}+c

Option D is correct because the integral can be rewritten in the form

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

, which simplifies to

e^x f(x) + c

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

706 Easy

706 of 746

Understanding +4 | -1

If polynomials

P

and

Q

satisfies

\int((3 x-1) \cos x+(1-2 x) \sin x) d x

=P \cos x+Q \sin x

(ignoring the constant of integration) then

View answer and explanation

Correct option: D

Q=3(x-1)

Option D (

Q=3(x-1)

) is correct because after integrating by parts and comparing coefficients, Q is found to be 3(x-1).

707 Easy

707 of 746

Application +4 | -1

The value of

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

, is

View answer and explanation

Correct option: B

\frac{2}{5}\left(x+\frac{1}{x}\right)^{5 / 2}+C

Option B is correct because the integral simplifies to (2/5)(x + 1/x)^(5/2) + C after recognizing the derivative pattern.

708 Easy

708 of 746

Understanding +4 | -1

\int \frac{{dx}}{{x\,\,\log \,\,x}} = f\left( x \right) +

constant, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: C

\log \,\,\log \,\,x

Option C (

\log \, \log \, x

) is correct because the integral of

\frac{1}{x \, \log x}

\log(\log x)

709 Easy

709 of 746

Application +4 | -1

\int \cos \sqrt{x} d x=

View answer and explanation

Correct option: A

2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]+c

Option A (

2[\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x}]+c

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

710 Easy

710 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

2\,\sqrt {\tan \,\,x} \,\, + \,\,C

Option A (

2\,\sqrt {\tan \,x} \, + \,C

) is correct because it is the result of a direct u-substitution and integration.

711 Easy

711 of 746

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\left\{x-\frac{1}{2 \sqrt{2}} \log \left|\frac{\sqrt{2}+\sin 2 x}{\sqrt{2}-\sin 2 x}\right|\right\}+C

The correct answer is C (

\frac{1}{2}\left\{x-\frac{1}{2 \sqrt{2}} \log \left|\frac{\sqrt{2}+\sin 2 x}{\sqrt{2}-\sin 2 x}\right|\right\}+C

) because after applying trigonometric identities and simplification, the integral can be evaluated.

712 Easy

712 of 746

Application +4 | -1

\int \frac{\mathrm{d} x}{4-3 \cos ^2 x+5 \sin ^2 x}=\frac{1}{3} f(3 \tan x)+C

, then

f(x)

is equal to

View answer and explanation

Correct option: D

\tan ^{-1} x

Option D (\tan^-1 x\\) is correct. Dividing by \cos² x and substituting \

t = \tan x

simplifies the integral to a standard arctangent form.

713 Easy

713 of 746

Analyzing +4 | -1

\int \frac{1}{{(\sin \,\,x\,\, + \,\,4)\,(\sin \,\,x - 1)}}\,\,dx

= A\frac{1}{{\tan \,\,\frac{x}{2} - 1}} + B{\tan ^{ - 1}}\left( {f\left( x \right)} \right) + C

, then

View answer and explanation

Correct option: D

A = \frac{2}{5},\;\,B = \frac{{ - \,2}}{{5\,\sqrt {15} }},\;\,f\left( x \right) = \frac{{4\tan \,\,x/2 + 1}}{{\sqrt {15} }}

Option D (

A = \frac{2}{5},\\;B = \frac{{ - \\,2}}{{5\\;\sqrt {15} }},\\;\,f\left( x \right) = \frac{{4\tan \\,\frac{x}{2}\\ + \\,1}}{{\sqrt {15} }}

) is correct because it represents the correct partial fraction decomposition and integration of the given expression.

714 Easy

714 of 746

Application +4 | -1

Let

\int \,\,{e^x}\,\,\left( {f\,\,\left( x \right) - f'\left( x \right)} \right)\,\,dx = \phi \,\,\left( x \right)

Then,

\int \,\,{e^x}\,f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\,\,\left[ {\phi \left( x \right)\,\, + \,\,{e^x}f\,\,\left( x \right)} \right]

Option C (

\frac{1}{2}\,\left[ {\phi \left( x \right)\, + \,{e^x}f\,\,\left( x \right)} \right]

) is correct because it is derived from integration by parts and the given equation.

715 Easy

715 of 746

Application +4 | -1

The integral

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

is equl to

View answer and explanation

Correct option: B

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

Option B is correct because the integral simplifies to a form that can be solved by substitution, resulting in the given expression.

716 Easy

716 of 746

Analyzing +4 | -1

Integral of

\sqrt{1+2 \cot x(\cot x+\operatorname{cosec} x)}

w.r.t.

x

, is

View answer and explanation

Correct option: B

2 \ln \sin \frac{x}{2}+C

Option B (

2 \ln \sin \frac{x}{2}+C

) is correct because simplifying the expression inside the square root leads to an integral that can be directly evaluated to this result.

717 Easy

717 of 746

Application +4 | -1

f(x)=\frac{x^2}{1+x^2}

and

g(x)=\sin x

then

\int f o g(x) \cos x \mathrm{~d} x

View answer and explanation

Correct option: A

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

Option A (\sin x - \tan^-1(\sin x) + C\\) is correct. By substituting \

t = \sin x

, the integral simplifies to \

\int \frac{t^2}{1+t^2} dt

, which can be easily integrated.

718 Easy

718 of 746

Application +4 | -1

\int \,\,\frac{{1\,\, + \,\,x}}{{1\,\, + \,\,\sqrt[3]{x}}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{3}{5}\,\,{x^{5/3}}\,\, - \,\,\frac{3}{4}{x^{4/3}}\,\, + \,\,x\,\, + \,\,C

Option A (

\frac{3}{5}\cdot{x^{5/3}}\\ - \frac{3}{4}{x^{4/3}}\\ + \\ x\\ + \\ C

) is correct because after integrating the simplified expression, we obtain this result.

719 Easy

719 of 746

Application +4 | -1

\int \frac{{\sec \,\,x}}{{\sec \,\,x\,\, + \,\,\tan x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: A

\tan \,\,x - \sec \,\,x + c

Option A (

\tan \,x - \sec \,x + c

) is correct because it is obtained by multiplying the numerator and denominator by

(\sec x - \tan x)

and then integrating the resulting expression.

720 Easy

720 of 746

Application +4 | -1

The integral

\int \frac{d x}{\sqrt{x}+\sqrt[3]{x^2}}

represents the function

View answer and explanation

Correct option: B

3 \sqrt[3]{x}-6 \sqrt[6]{x}+6 \log |1+\sqrt[6]{x}|+c

Option B (

3 \sqrt[3]{x}-6 \sqrt[6]{x}+6 \log |1+\sqrt[6]{x}|+c

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

Revision notes and key takeaways

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