$\smallint {\cos ^3}\,\,x\,.\,{e^{\log \sin x}}\,\,dx$ is equal to

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

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

\[\smallint \,\,\frac{{{a^{x/2}}}}{{\sqrt {{a^{ - x}} - \,\,{a^x}} }}\,\,dx = \]

$\frac{1}{{\log \,\,a}}\,\,{\tan ^{ - 1}}\,\,\left( {{a^x}} \right)$

$\log \,\,\left( {{a^x} - \,\,1} \right)$

$\smallint \frac{{dx}}{{\sin \,\,x - \cos \,\,x + \sqrt 2 }}$ equals

$ - \frac{1}{{\sqrt 2 }}\,\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right) + c\;$

$\frac{1}{2}\,\,\tan \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right) + c$

The value of the integral $\smallint \,\,\frac{{1 + {x^2}}}{{1 + {x^4}}}\,\,dx$ is equal to

${\tan ^{ - 1}}\,\,{x^2}\, + \,\,C$

$\frac{1}{{2\,\sqrt 2 }}\,\,\log \,\,\left( {\frac{{{x^2}\,\, + \sqrt {\,\,2} x\,\, + \,\,1}}{{{x^2}\,\, - \,\,\sqrt 2 x\,\, + \,\,1}}} \right) + C$

$\smallint \,\,\frac{{{4^{x + 1}}\,\, - \,\,{7^{x - 1}}}}{{{{28}^x}}}\;\,\,dx$ is equal to

$\frac{{{4^{ - x}}}}{{{{\log }_e}\,\,7}}\,\, - \,\,\frac{{{7^{ - x}}}}{{{{\log }_e}\,\,4}}\,\, + \,\,c$

$\frac{{{4^{ - x}}}}{{{{\log }_e}\,\,4}}\,\, - \,\,\frac{{{7^{ - x}}}}{{{{\log }_e}\,\,7}}\,\, + \,\,c$

If a function $f\left( x \right)$ satisfies $f'\left( x \right) = g\left( x \right)$ Then, the value of $\mathop \smallint \nolimits_a^b \,\,f\left( x \right)\,\,g\,\,\left( x \right)\,\,dx$ is

$\frac{1}{2}\,\,{\left[ {\,f\left( b \right)\,\, - \,\,f\left( a \right)} \right]^2}$

The value of $\int \log (1-\sqrt{x}) \mathrm{d} x$ is

$x \log (1-\sqrt{x})-\frac{1}{2} x-\frac{3}{2} \sqrt{x}+C$

$x^2 \log (1-\sqrt{x})-\frac{1}{2} \sqrt{x}+C$

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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 721-746 of 746 questions

721 Easy

721 of 746

Application +4 | -1

\frac{d}{d x} f(x)=x \cos x+\sin x

and

f(0)=2

, then

f(x)=

View answer and explanation

Correct option: C

x \sin x+2

Option C is correct because integrating x cos(x) + sin(x) results in x sin(x) + C, and using the initial condition f(0) = 2, we find that C = 2, so f(x) = x sin(x) + 2.

722 Easy

722 of 746

Application +4 | -1

\int {\cos ^3}\,\,x\,.\,{e^{\log \sin x}}\,\,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 because after simplifying the expression and using substitution, the integral evaluates to this result.

723 Easy

723 of 746

Application +4 | -1

\int 5^{\log _{\mathrm{e}} \mathrm{x}} \mathrm{dx}=

View answer and explanation

Correct option: B

\frac{\mathrm{x}^{\left(\log _c 5+1\right)}}{\log _e 5+1}+\mathrm{c}

Option B (

\frac{{\mathrm{x}^{\left(\log _c 5+1\right)}}}{{\log _e 5+1}}+\mathrm{c}

) is correct because

5^{\log_e x} = x^{\log_e 5}

, and integrating

x^{\log_e 5}

gives the result.

724 Easy

724 of 746

Application +4 | -1

The value of the integral

\int \frac{x \sin x^2 e^{\sec x^2}}{\cos ^2 x^2} d x

, is

View answer and explanation

Correct option: A

\frac{1}{2} e^{\sec x^2}+C

Option A is correct because the integral is solved by recognizing the derivative of the secant term and applying a simple substitution.

725 Easy

725 of 746

Application +4 | -1

\int \,\,\frac{{{a^{x/2}}}}{{\sqrt {{a^{ - x}} - \,\,{a^x}} }}\,\,dx =

View answer and explanation

Correct option: A

\frac{1}{{\log \,\,a}}\,\,{\sin ^{ - 1}}\,\,\left( {{a^x}} \right)

Option A (

\frac{1}{{\log \,a}}\,{\sin ^{ - 1}}\,\left( {{a^x}} \right)

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

726 Easy

726 of 746

Application +4 | -1

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

View answer and explanation

Correct option: D

\log (\sec x+\tan x)-\tan \frac{x}{2}+c

Option D (

\log (\sec x+\tan x)-\tan \frac{x}{2}+c

) is correct because the integral is solved using trigonometric identities and simplification.

727 Easy

727 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A 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+1}

and

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

728 Easy

728 of 746

Application +4 | -1

\int \frac{2^{x+1}-5^{x-1}}{10^x} d x

is equal to

View answer and explanation

Correct option: B

\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c

Option B (

\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c

) is correct because it is the result of the integration after simplifying the expression using exponent rules.

729 Easy

729 of 746

Understanding +4 | -1

\int f(x) d x=f(x)

, then

View answer and explanation

Correct option: D

f(x)=e^x

Option D (

f(x)=e^x

) is correct because the integral of

e^x

e^x

(plus a constant, which we can ignore in this context).

730 Easy

730 of 746

Application +4 | -1

\int \frac{{dx}}{{\sin \,\,x - \cos \,\,x + \sqrt 2 }}

equals

View answer and explanation

Correct option: D

- \frac{1}{{\sqrt 2 }}\,\,\cot \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right) + c

The correct answer is D (

- \frac{1}{{\sqrt 2 }}\,\,\cot \,\,\left( {\frac{x}{2}\,\, + \,\,\frac{\pi }{8}} \right) + c

) because after applying trigonometric identities and integrating, the result matches option D.

731 Easy

731 of 746

Understanding +4 | -1

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

View answer and explanation

Correct option: B

\log \left(1+\sin ^2 x\right)+c

Option B (

\log \left(1+\sin ^2 x\right)+c

) is correct because the derivative of

1+\sin^2 x

\sin 2x

, which is present in the integrand.

732 Easy

732 of 746

Analyzing +4 | -1

The value of

\int \frac{x^2-1}{\left(x^2+1\right) \sqrt{x^4+1}} d x

, is equal to

View answer and explanation

Correct option: A

\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^2+1}{\sqrt{2} x}\right)+C

Option A is correct because after dividing by

x^2

and substituting

x + \frac{1}{x} = t

, the integral simplifies to

\frac{1}{\sqrt{2}} \sec ^{-1}\left(\frac{x^2+1}{\sqrt{2} x}\right)+C

733 Easy

733 of 746

Analyzing +4 | -1

\int \frac{e^{\left(x^2+4 \ln x\right)}-x^3 e^{x^2}}{x-1} d x

is equal to

View answer and explanation

Correct option: D

None of these

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

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

, which is not any of the given options.

734 Easy

734 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

-\frac{1}{a} \sqrt{\frac{x+a}{x-a}}+c

Option D (

-\frac{1}{a} \sqrt{\frac{x+a}{x-a}}+c

) is correct because after substitution and integration, the expression simplifies to this form.

735 Easy

735 of 746

Analyzing +4 | -1

\int \frac{x^2}{(x \sin x+\cos x)^2} d x

would be equal to

View answer and explanation

Correct option: B

\frac{\sin x-x \cos x}{x \sin x+\cos x}+C

Option B is correct because after applying integration by parts and simplifying, we get

\frac{\sin x-x \cos x}{x \sin x+\cos x}+C

Explore Other Chapters in MATHEMATICS

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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

736 Easy

736 of 746

Application +4 | -1

The value of the integral

\int \,\,\frac{{1 + {x^2}}}{{1 + {x^4}}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{1}{{\sqrt 2 }}\,\,{\tan ^{ - 1}}\,\,\left( {\frac{{{x^2} - \,\,1}}{{\sqrt 2 x}}} \right)

Option B (

\frac{1}{{\sqrt 2 }}\\,{\tan ^{ - 1}}\left( {\frac{{{{{x^2}} - \\,1}}}{{\sqrt 2 x}}} \right)

) is correct because the integral is solved by dividing numerator and denominator by

x^2

and then using a suitable substitution.

737 Easy

737 of 746

Application +4 | -1

\int \frac{\tan (\log x)}{x} d x=

View answer and explanation

Correct option: C

\log \sec (\log x)+c

Option C is correct because substituting

t = \log x

transforms the integral to

\int \tan t dt

, which equals

\log |\sec t| + c = \log |\sec(\log x)| + c

738 Easy

738 of 746

Application +4 | -1

\int \frac{x^7}{\left(1+x^4\right)^2} \mathrm{~d} x=\frac{1}{4}\left[\log \left(1+x^4\right)+f(x)\right]+C

then

View answer and explanation

Correct option: C

f(x)=\frac{1}{1+x^4}

The correct answer is

f(x)=\frac{1}{1+x^4}

because after substituting

t=1+x^4

and integrating, we can identify

f(x)

by comparing with the given expression.

739 Easy

739 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{-e^{-x}}{e^x+e^{-x}}+C

Option A (

\frac{-e^{-x}}{e^x+e^{-x}}+C

) is correct because after simplifying the integral, the result matches this option.

740 Easy

740 of 746

Application +4 | -1

Let

\int e^x\left(f(x)-f^{\prime}(x)\right) d x=\phi(x)

, then

\int e^x f(x) d x

is equal to

View answer and explanation

Correct option: C

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

Option C is correct because by integrating

e^x(f(x) + f'(x))

and

e^x(f(x) - f'(x))

and adding the results, we can isolate the integral of

e^x f(x)

741 Easy

741 of 746

Analyzing +4 | -1

The value of

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

, is equal to

View answer and explanation

Correct option: B

\log \left|\frac{x e^x}{1+x e^x}\right|+\frac{1}{1+x e^x}+C

Option B is correct because after substituting

1+xe^x = t

and using partial fractions, the integral evaluates to

\log \left|\frac{x e^x}{1+x e^x}\right|+\frac{1}{1+x e^x}+C

742 Easy

742 of 746

Application +4 | -1

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

is equal to:

View answer and explanation

Correct option: D

x^x \ln x+C

Option D (

x^x \ln x+C

) is correct because it results from correctly identifying the substitution and integrating the expression.

743 Easy

743 of 746

Application +4 | -1

\int \,\,\frac{{{4^{x + 1}}\,\, - \,\,{7^{x - 1}}}}{{{{28}^x}}}\;\,\,dx

is equal to

View answer and explanation

Correct option: A

\frac{1}{{7\;\,\,{{\log }_e}4}}\,\,{4^{ - x}}\,\, - \,\,\frac{4}{{{{\log }_e}7}}\,\,{7^{ - x}}\,\, + \,\,c

Option A (

\frac{1}{{7\;\,\,{\log }_e4}}\,{4^{ - x}}\, - \,\frac{4}{{{\log }_e7}}\,{7^{ - x}}\, + \,c

) is correct because it is the result of integrating the given expression after simplification.

744 Easy

744 of 746

Analyzing +4 | -1

\int \sin ^5 x \cos ^4 x \,d x=A \cos ^9 x+B \cos ^7 x+C \cos ^5 x+D,

then

9 A+7 B+5 C=

View answer and explanation

Correct option: B

Option B is correct because after evaluating the integral and comparing coefficients, we find that

9A + 7B + 5C = 0

745 Easy

745 of 746

Application +4 | -1

If a function

f\left( x \right)

satisfies

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

Then, the value of

\int \nolimits_a^b \,\,f\left( x \right)\,\,g\,\,\left( x \right)\,\,dx

View answer and explanation

Correct option: A

\frac{1}{2}\,\,\left[ {{{\left( {\,f\left( b \right)} \right)}^2}\,\, - \,\,{{\left( {\,F\left( a \right)} \right)}^2}} \right]

Option A is correct because the integral of f(x)g(x) from a to b, where f'(x) = g(x), evaluates to (1/2)[(f(b))² - (f(a))²].

746 Easy

746 of 746

Application +4 | -1

The value of

\int \log (1-\sqrt{x}) \mathrm{d} x

View answer and explanation

Correct option: B

(x-1) \log (1-\sqrt{x})-\frac{1}{2} x-\sqrt{x}+C

Option B is correct because after integrating by parts and simplifying, the value of k is found to be

\frac{5 \pi}{4}

Revision notes and key takeaways

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