The value of the integral $\mathop \smallint \limits_0^1 \log \,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right)\,\,dx$ , is

$ - \frac{\pi }{2}\,\,\log \,\,2$

The value of $\int \frac{\mathrm{d} x}{5+4 \cos x}$ is

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

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

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

The value of $\smallint \,\,\frac{{{e^{5{{\log }_e}x}}\,\, - \,\,{e^{4{{\log }_e}x}}}}{{{e^{3{{\log }_e}x}}\,\, - \,\,{e^{2{{\log }_e}x}}}}\;\,\,dx$ is

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

$\smallint \,\,\frac{{\sec \,\,x\,\,{\rm{cosec}}\,\,x}}{{2\,\,\cot \,\,x - \,\,\sec x\,\,{\rm{cosec}}\,\,x}}\,\,dx$ is equal to

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

$\log \,\,|\sec \,\,x\,\, + \,\,{\rm{cosec}}\,\,x| + \,\,c$

$\log \,\,|\sec \,\,2x\,\, + \,\,{\rm{cosec}}\,\,2x| + \,\,c$

$\smallint \,\,\frac{{{{({{\tan }^{ - 1}}\,\,x)}^3}}}{{1\,\, + \,\,{x^2}}}\;\,\,dx$ is equal to

$3\;{({\tan ^{ - 1}}\,\,x)^2} + \,\,c$

$\smallint {\cos ^3}\,\,x{e^{\log \left( {\sin x} \right)}}\,\,dx$ is equal to

$ - \,\,\frac{{{{\sin }^4}\,\,x}}{4}\,\, + \,\,C$

$\frac{{{e^{\sin x}}}}{4}\,\, + \,\,C$

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

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

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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 661-690 of 746 questions

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661 Easy

661 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

2 \sqrt{1+\sqrt{1+x^2}}+C

Option D is correct because the integral simplifies to

2 \sqrt{1+\sqrt{1+x^2}}+C

after a suitable substitution.

662 Easy

662 of 746

Analyzing +4 | -1

The value of the integral

\int \limits_0^1 \log \,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right)\,\,dx

, is

View answer and explanation

Correct option: B

- \,\log \,\,2

The correct answer is B (

- \,\log \,\,2

) because after substitution, the integral can be solved using a known result.

663 Easy

663 of 746

Application +4 | -1

The value of

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

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the result of integrating the given expression using the tangent half-angle substitution.

664 Easy

664 of 746

Application +4 | -1

\int x^3 \sqrt{3+5 x^4} d x=

View answer and explanation

Correct option: C

\frac{1}{30}\left(3+5 x^4\right)^{3 / 2}+c

Option C is correct because the integral evaluates to

\frac{1}{30}(3 + 5x^4)^{3/2} + c

after applying the substitution method.

665 Easy

665 of 746

Application +4 | -1

\int x^{-6}\left(1+2 x^3\right)^{2 / 3} d x

is equal to:

View answer and explanation

Correct option: A

-\frac{1}{5}\left(x^{-3}+2\right)^{5 / 3}+c

Option A (

-\frac{1}{5}\left(x^{-3}+2\right)^{5 / 3}+c

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

666 Easy

666 of 746

Application +4 | -1

The value of

\int \,\,\frac{{{e^{5{{\log }_e}x}}\,\, - \,\,{e^{4{{\log }_e}x}}}}{{{e^{3{{\log }_e}x}}\,\, - \,\,{e^{2{{\log }_e}x}}}}\;\,\,dx

View answer and explanation

Correct option: C

\frac{{{x^3}}}{3}\,\, + \,\,c

Option C (

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

) is correct because it is the result of simplifying the expression and then integrating.

667 Easy

667 of 746

Analyzing +4 | -1

\int \,\,\frac{{\sec \,\,x\,\,\mathrm{cosec}\,\,x}}{{2\,\,\cot \,\,x - \,\,\sec x\,\,\mathrm{cosec}\,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\,\,\log \,\,|\sec \,\,2x\,\, + \,\,\tan \,\,2x| + \,\,c

Option C is correct because the integral simplifies to the integral of sec(2x), which integrates to (1/2)log|sec(2x) + tan(2x)| + c.

668 Easy

668 of 746

Application +4 | -1

\int \,\,\frac{{{{({{\tan }^{ - 1}}\,\,x)}^3}}}{{1\,\, + \,\,{x^2}}}\;\,\,dx

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

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

669 Easy

669 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

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

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

670 Easy

670 of 746

Application +4 | -1

The value of

\int {\frac{{{x^3}}}{{1 + {x^8}}}dx}

is equal to

View answer and explanation

Correct option: A

\frac{1}{4}{\tan ^{ - 1}}{x^4} + c

Option A (

\frac{1}{4}{\tan ^{ - 1}}{x^4} + c

) is correct because after a simple substitution, the integral transforms into a standard arctangent integral.

671 Easy

671 of 746

Understanding +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B ((((tan^-1x)⁴)/4) + c) is correct because it is the result of a straightforward u-substitution where u = tan^-1x.

672 Easy

672 of 746

Application +4 | -1

\int \frac{d x}{(x+1)(x+2)}=

View answer and explanation

Correct option: C

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

Option C,

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

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

673 Easy

673 of 746

Application +4 | -1

\int \frac{\sqrt{1+\sqrt[3]{x}}}{x^{2 / 3}} \mathrm{~d} x=2 f(x)^{3 / 2}+C

then

f(x)

is equal to

View answer and explanation

Correct option: B

1+x^{1 / 3}

Option B (

1+x^{1 / 3}

) is correct because after substituting

t^2 = 1 + x^{1/3}

, the integral simplifies to

2(1+x^{1/3})^{3/2} + C

, matching the given form.

674 Easy

674 of 746

Application +4 | -1

\int {\cos ^3}\,\,x{e^{\log \left( {\sin x} \right)}}\,\,dx

is equal to

View answer and explanation

Correct option: B

- \,\,\frac{{{{\cos }^4}\,\,x}}{4}\,\, + \,\,C

Option B (

\ - \,\,\frac{{{{\cos }^4}\,x}}{4}\, + \,C

) is correct. Simplifying the integral using properties of logarithms and then integrating leads to the correct result.

675 Easy

675 of 746

Application +4 | -1

\int \,\,\frac{{{x^3}\,\,\sin \,\,\left[ {{{\tan }^{ - 1}}\,\,\left( {{x^4}} \right)} \right]}}{{1\,\, + \,\,{x^8}}}\,\,dx

is equal to

View answer and explanation

Correct option: C

- \frac{1}{4}\,\,\cos \,\,\left[ {{{\tan }^{ - 1}}\,\,\left( {{x^4}} \right)} \right]\,\, + \,\,c

Option C (

- \frac{1}{4}\cos \left[ {{{\tan }^{ - 1}}\left( {{x^4}} \right)} \right]\, + \,c

) is correct because it is the result of the given integral after applying the correct substitutions and integration.

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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

676 Easy

676 of 746

Application +4 | -1

\int \frac{d x}{\sin ^5 x \cos ^5 x}

is equal to

View answer and explanation

Correct option: B

\frac{1}{4}\left(\tan ^4 x-\cot ^4 x\right)+2\left(\tan ^2 x-\cot ^2 x\right)+6 \log |\tan x|

+C

Option B is correct because the integral evaluates to

\frac{1}{4}(\tan ^4 x-\cot ^4 x)+2(\tan ^2 x-\cot ^2 x)+6 \log |\tan x| +C

after simplification.

677 Easy

677 of 746

Application +4 | -1

The value of

\int \,\,\frac{{dx}}{{{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}

View answer and explanation

Correct option: A

- \,\,\frac{{{{\left( {{x^4} + 1} \right)}^{1/4}}}}{x}\,\, + \,\,c

Option A (

- \,\,\frac{{{{\left( {{x^4} + 1} \right)}^{1/4}}}}{x}\, + \,c

) is correct because the integral is solved using a substitution and simplification.

678 Easy

678 of 746

Application +4 | -1

The value of

\sqrt{2} \int \frac{\sin x}{\sin \left(x-\frac{\pi}{4}\right)} d x

, is

View answer and explanation

Correct option: A

x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+C

Option A (

x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+C

) is correct because after applying trigonometric identities and integrating, the result matches this option.

679 Easy

679 of 746

Application +4 | -1

\int \frac{x^2-1}{x \sqrt{x^4+3 x^2+1}} d x=\log _e(f(x))+c

where

f (x)

is given by

View answer and explanation

Correct option: D

\frac{1}{x}\left(x^2+1+\sqrt{x^4+3 x^2+1}\right)

Option D (

\frac{1}{x}\left(x^2+1+\sqrt{x^4+3 x^2+1}\right)

) is correct because after simplifying and integrating, the result matches the given form with this f(x).

680 Easy

680 of 746

Application +4 | -1

The value of

\int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x d x

is equal to

View answer and explanation

Correct option: C

\frac{5^{5^{5^x}}}{(\ell n 5)^3}+C

Option C (

\frac{5^{5^{5^x}}}{(\ell n 5)^3}+C

) is correct because it is the result of repeated u-substitutions and integration of an exponential function.

681 Easy

681 of 746

Application +4 | -1

\int \log _{10} x d x=

View answer and explanation

Correct option: D

x\left(\log _{10} x-\log _{10} e\right)+c

Option D (

x(\log _{10} x-\log _{10} e)+c

) is correct because it is the result of integrating

\log_{10} x

using integration by parts and the change of base formula.

682 Easy

682 of 746

Analyzing +4 | -1

\int \frac{\sin ^3 x}{\left(\cos ^4 x+3 \cos ^2 x+1\right) \tan ^{-1}(\sec x+\cos x)} d x=

View answer and explanation

Correct option: B

\log _e\left|\tan ^{-1}(\sec x+\cos x)\right|+C

The correct answer is B (

\log _e\left|\tan ^{-1}(\sec x+\cos x)\right|+C

) because the integral simplifies to a form where the derivative of the inner function is present, allowing for a direct integration.

683 Easy

683 of 746

Analyzing +4 | -1

\int {\frac{{dx}}{{{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {\frac{{x - 1}}{3}} \right) + \frac{{f(x)}}{{{x^3} - 2x + 10}}} \right) + C,

where, C is a constant of integration, then

View answer and explanation

Correct option: C

A = \frac{1}{{54}}\;\,\,\mathrm{and}\,\;f(x) = 3(x - 1)

Option C (

A = \frac{1}{{54}}

and

f(x) = 3(x - 1)

) is correct because it matches the result obtained after integrating the given expression and comparing coefficients.

684 Easy

684 of 746

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\frac{1}{2} \log _e\left(\tan ^2 x+\sqrt{1+\tan ^4 x}\right)+C

The correct answer is B (

\frac{1}{2} \log _e\left(\tan ^2 x+\sqrt{1+\tan ^4 x}\right)+C

) because after simplification and substitution, the integral matches the form of the integral of

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

685 Easy

685 of 746

Application +4 | -1

\int\{1+2 \tan x(\tan x+\sec x)\}^{\frac{1}{2}} d x

is equal to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the integral simplifies to

\int (\sec x + \tan x) dx

, which evaluates to

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

686 Easy

686 of 746

Analyzing +4 | -1

f(x) = \int {\frac{{5{x^8} + 7{x^6}}}{{{{({x^2} + 1 + 2{x^7})}^2}}}dx,(x \ge 0)}

, and

f(0) = 0

, then the value of

f (1)

View answer and explanation

Correct option: C

\frac{1}{4}

Option C (

\frac{1}{4}

) is correct because after evaluating the integral and applying the initial condition f(0) = 0, we find that f(1) = 1/4.

687 Easy

687 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\log \,\,|\tan \,\,x| + \,c

Option B (

\log \\,|\tan \\,x| + \\,c

) is correct because the integral can be simplified using trigonometric identities and then solved by recognizing the derivative of

\tan x

in the numerator.

688 Easy

688 of 746

Application +4 | -1

\int \frac{1}{x-x^3} d x=

View answer and explanation

Correct option: D

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

Option D,

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

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

689 Easy

689 of 746

Application +4 | -1

\int \tan ^4 x d x=

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral simplifies to this form after using trigonometric identities and integration.

690 Easy

690 of 746

Application +4 | -1

The integral of

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

View answer and explanation

Correct option: C

\frac{x}{\ln x}+c

Option C (

\frac{x}{\ln x}+c

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

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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