$\smallint \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\sqrt {{x^2} - \,\,1} }}\;\,\,dx$ is equal to

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

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

$\smallint \frac{{{{\left( {{x^4} - \,\,x} \right)}^{1/4}}}}{{{x^5}}}\,\,dx$ is equal to

$\frac{4}{5}\,\,{\left( {1\,\, - \,\,\frac{1}{{{x^3}}}} \right)^{5/4}} + \,\,C$

$\smallint \,\,\frac{{\cos \,\,x\,\, + \,\,x\,\,\sin \,\,x}}{{x\,\,\left( {x\,\, + \,\,\cos \,\,x} \right)}}\,\,dx$ is equal to

$\log \,\,\left( {x\,\,\left( {x\,\, + \,\,\cos \,\,x} \right)} \right) + C$

$\log \,\,\left( {\frac{{x + \,\,\cos \,\,x}}{x}} \right)$

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

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

$\smallint \,\,\frac{1}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}\,\,dx$ is equal to

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\cot \,\,\frac{x}{2}\,} \right| + C$

$\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, - \,\,\frac{\pi }{8}} \right)\,} \right| + C$

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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 901-930 of 995 questions

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901 Easy

901 of 995

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.

902 Easy

902 of 995

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.

903 Easy

903 of 995

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).

904 Medium

904 of 995

Application +4 | -1

\int \frac{1}{x \sqrt{1+\log x}} d x=

View answer and explanation

Correct option: C

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

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

905 Easy

905 of 995

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.

906 Medium

906 of 995

Application +4 | -1

\int \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\sqrt {{x^2} - \,\,1} }}\;\,\,dx

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because after trigonometric substitution and simplification, the result matches this option.

907 Medium

907 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A is correct because the integral can be solved by rewriting the integrand and using substitution, leading to the expression

x - \log[1 + \sqrt{1 - e^{2x}}]+c

908 Medium

908 of 995

Application +4 | -1

\int \frac{{{{\left( {{x^4} - \,\,x} \right)}^{1/4}}}}{{{x^5}}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{4}{{15}}\,\,{\left( {1 - \,\,\frac{1}{{{x^3}}}} \right)^{5/4}} + \,\,C

Option A is correct because the integral simplifies to a form where the power rule can be applied after a suitable substitution.

909 Medium

909 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C is correct because the integral simplifies to

\frac{1}{2} \int e^{2x} \sec^2 x dx + \int e^{2x} \tan x dx

, which integrates to

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

910 Easy

910 of 995

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.

911 Hard

911 of 995

Analyzing +4 | -1

\int \frac{e^{\sec x} \sin [x+(\pi / 4)]}{\cos x(1-\sin x)} d x\left(0 < x < \frac{\pi}{4}\right)

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because after simplification and integration, the integral evaluates to

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

912 Easy

912 of 995

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.

913 Easy

913 of 995

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}}

914 Medium

914 of 995

Understanding +4 | -1

\int \,\,\frac{{\cos \,\,x\,\, + \,\,x\,\,\sin \,\,x}}{{x\,\,\left( {x\,\, + \,\,\cos \,\,x} \right)}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\log \,\,\left( {\frac{x}{{x\,\, + \,\,\cos \,\,x}}} \right) + C

Option B (

\log \,\,\left( {\frac{x}{{x\,\, + \,\,\cos \,\,x}}} \right) + C

) is correct because the integral can be simplified and solved using basic integration rules.

915 Easy

915 of 995

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.

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916 Easy

916 of 995

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.

917 Medium

917 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because by substituting

t=e^x

and then

t=\tan \theta

, we can evaluate the integral.

918 Easy

918 of 995

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.

919 Easy

919 of 995

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.

920 Easy

920 of 995

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}

921 Easy

921 of 995

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.

922 Easy

922 of 995

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

923 Medium

923 of 995

Application +4 | -1

\int \,\,\frac{1}{{\cos \,\,x\,\, - \,\,\sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{{\sqrt 2 }}\,\,\log \,\,\left| {\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{{3\pi }}{8}} \right)\,} \right| + C

Option A (

\frac{1}{{\sqrt 2 }}\,\log \,\left| {\,\tan \,\left( {\frac{x}{2}\, + \\,\frac{{3\pi }}{8}} \right)\} \right| + C

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

924 Easy

924 of 995

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

925 Easy

925 of 995

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.

926 Medium

926 of 995

Application +4 | -1

I(x)=\int e^{\sin ^2 x}(\cos x \sin 2 x-\sin x) d x

and

I(0)=

1 , then

I\left(\frac{\pi}{3}\right)

is equal to

View answer and explanation

Correct option: B

\frac{1}{2} e^{\frac{3}{4}}

Option B (

\frac{1}{2} e^{\frac{3}{4}}

) is correct because it involves solving the given integral and applying the initial condition.

927 Medium

927 of 995

Application +4 | -1

k\,\int \nolimits_0^1 \,\,x\,.\,f\left( {3x} \right)dx = \int \nolimits_0^3 \,t\,.\,f\,\,\left( t \right)\,\,dt

then the value of

k

View answer and explanation

Correct option: A

Option A (9) is correct because by substituting

t = 3x

, the integral transforms, and solving for

k

yields 9.

928 Easy

928 of 995

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.

929 Easy

929 of 995

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)

930 Easy

930 of 995

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.

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Review INDEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

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