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$

$\smallint \,\,\sqrt {\frac{{\cos \,\,x\,\, - \,\,{{\cos }^3}\,x}}{{1\,\, - \,\,{{\cos }^3}\,\,x}}} \,\,dx$ is equal to

$\frac{2}{3}\,\,{\sin ^{ - 1}}\left( {{{\cos }^{3/2}}\,\,x} \right) + C$

$\frac{3}{2}\,\,{\sin ^{ - 1}}\left( {{{\cos }^{3/2}}\,\,x} \right) + C$

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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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 931-960 of 995 questions

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931 Easy

931 of 995

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.

932 Easy

932 of 995

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.

933 Easy

933 of 995

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

934 Easy

934 of 995

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.

935 Medium

935 of 995

Analyzing +4 | -1

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

equals

View answer and explanation

Correct option: B

\frac{1}{\sqrt{3}} \operatorname{Tan}^{-1} \frac{(\tan x-\cot x)}{\sqrt{3}}+c

Option B is correct because the integral simplifies to an arctangent form after trigonometric manipulation and substitution.

936 Medium

936 of 995

Analyzing +4 | -1

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

View answer and explanation

Correct option: B

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

Option C (

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

) is correct because the integral can be expressed in the form

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

, where

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

and its derivative is

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

937 Medium

937 of 995

Application +4 | -1

The value of

\int \frac{x^7}{\left(1-x^2\right)^5} d x

, is

View answer and explanation

Correct option: B

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

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

\frac{1}{8} \frac{x^8}{(1-x^2)^4}+C

, but the explanation provided uses a substitution that changes the variable to theta and then back to x, which is not necessary.

938 Medium

938 of 995

Understanding +4 | -1

f_n(x)=\log \log \ldots \log x

(log is repeated

n

times) then

\int\left[x f_1(x) f_2(x) \ldots f_n(x)\right]^{-1} \mathrm{~d} x

is equal to

View answer and explanation

Correct option: A

f_{n+1}(x)+C

Option A (

f_{n+1}(x)+C

) is correct because it is the result of applying the substitution method and integrating.

939 Easy

939 of 995

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

940 Easy

940 of 995

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

941 Easy

941 of 995

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).

942 Easy

942 of 995

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.

943 Medium

943 of 995

Analyzing +4 | -1

\int e^{\sec x} \cdot \sec ^3 x \left(\sin ^2 x+\cos x+\sin x+\sin x \cos x\right) d x=

View answer and explanation

Correct option: A

e^{\sec x}(\sec x+\tan x)+

The correct answer is A (

e^{\sec x}(\sec x+\tan x)+c

) because the integral simplifies to this expression after applying integration by parts and simplification.

944 Easy

944 of 995

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)

945 Hard

945 of 995

Application +4 | -1

Let

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

be a function satisfying

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

with

f\left( 0 \right) = 1

and

g\,\,\left( x \right)

be a function that satisfies

f\,\,\left( x \right) + g\left( x \right) = {x^2}

. Then, the value of the integral

\int \limits_0^1 \,f\left( x \right)\;\,\,g\,\,\left( x \right)\;\,dx

, is

View answer and explanation

Correct option: D

e\,\, - \,\,\frac{{{e^2}}}{2}\,\, - \,\,\frac{3}{2}

Option D (

e\,\, - \,\,\frac{{{{{e^2}}}}}{2}\,\, - \,\,\frac{3}{2}

) is correct because it is the result of evaluating the definite integral after finding the functions f(x) and g(x).

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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

946 Medium

946 of 995

Application +4 | -1

\int x \tan ^{-1} x d x=

View answer and explanation

Correct option: A

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

Option A is correct because the integral is solved using integration by parts, simplifying the resulting expression to match the given form.

947 Easy

947 of 995

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.

948 Medium

948 of 995

Analyzing +4 | -1

\int \,\,\sqrt {\frac{{\cos \,\,x\,\, - \,\,{{\cos }^3}\,x}}{{1\,\, - \,\,{{\cos }^3}\,\,x}}} \,\,dx

is equal to

View answer and explanation

Correct option: C

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

Option C (

\frac{2}{3}\\,{\cos ^{ - 1}}\\,\left( {{{{

\cos

}}^{3/2}}\\,x} \right) + C

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

949 Easy

949 of 995

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.

950 Easy

950 of 995

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.

951 Easy

951 of 995

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.

952 Medium

952 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the integral is solved using the form

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

953 Easy

953 of 995

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.

954 Medium

954 of 995

Application +4 | -1

The value of

\int x{\left( {{x^x}} \right)^x}\,\,\left( {2\,\,\log \,\,x\,\, + \,\,1} \right)\,\,dx

View answer and explanation

Correct option: A

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

Option A (

{\left( {{x^x}} \right)^x}\\ + \\ c

) is correct because the integral is solved by recognizing the derivative of (x^x)^x and using substitution.

955 Easy

955 of 995

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.

956 Easy

956 of 995

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.

957 Easy

957 of 995

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.

958 Easy

958 of 995

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.

959 Easy

959 of 995

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.

960 Easy

960 of 995

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.

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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