$\smallint \left( {\sqrt {\tan \,\,x} \,\, + \,\,\sqrt {\cot \,\,x} } \right)dx$ is equal to

$\sqrt 2 \,\,{\tan ^{ - 1}}\,(\sin \,\,x\,\, - \,\,\cos \,\,x)\,\, + \,\,c$

$\smallint {5^{{5^{{5^x}}}}} \cdot \,\,{5^{{5^x}}} \cdot \,\,{5^x}\;dx$ is equal to

$\frac{{{5^{{5^x}}}}}{{{{\left( {\log \,\,5} \right)}^3}}}\,\, + \,\,C$

The integral $\smallint \frac{{2x\,\, - \,\,3}}{{{{\left( {{x^2} + \,\,x\,\, + 1} \right)}^2}}}\,\,dx$ is equal to

$ - \,\,\frac{1}{{{x^2} + x + 1}}\,\, - \,\,\frac{4}{3}\,\,{\tan ^{ - 1}}\,\,\left( {4x + 3} \right) + C$

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

$\frac{1}{{4\left( {{x^2} + x + 1} \right)}}\,\, + \,\,\frac{2}{3}\,\,{\tan ^{ - 1}}\,\,\left( {2x + 1} \right) + C$

$\smallint \,\,{e^{x\log a}}\,{e^x}\;dx$ is equal to

$\frac{{{a^x}}}{{\log \,\,ae}}\,\, + \,\,c$

$\frac{{{e^x}}}{{1 + \,\,{{\log }_e}\,\,a}}\,\, + \,\,c$

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

If $\smallint \,\,\frac{1}{{x + {x^5}}}\,\,dx = f\left( x \right) + c$, then the value of $\smallint \frac{{{x^4}}}{{x + {x^5}}}\,\;dx$ equals

$f\left( x \right)\,\, + \,\,\log \,\,x\,\, + \,\,c$

$f\left( x \right)\,\, - \,\,\log \,\,x\,\, + \,\,c$

The value of the integral $\smallint \,\,{e^x}\,\,{\left( {\frac{{1 - x}}{{1 + x}}} \right)^2}\,dx$ is

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

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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691 Medium

691 of 995

Application +4 | -1

\int \left( {\sqrt {\tan \,\,x} \,\, + \,\,\sqrt {\cot \,\,x} } \right)dx

is equal to

View answer and explanation

Correct option: A

\sqrt 2 \,\,{\sin ^{ - 1}}\,(\sin \,\,x\,\, - \,\,\cos \,\,x)\,\, + \,\,c

Option A is correct because the integral simplifies to a form that can be solved using a trigonometric substitution, resulting in the given arcsine function.

692 Easy

692 of 995

Application +4 | -1

{I_n} = \int \,\,{x^n}\,.\,\,{e^{cx}}\;\,dx\;\;\,\mathrm{for}\,\;\;n\,\, \ge \,\,1,\,\;\mathrm{then}\,\,c\,.\,{I_n}\,\, + \,\,n\,.\,{I_{n - 1}}

is equal to

View answer and explanation

Correct option: A

{x^n}\,{e^{cx}}

Option A (

{x^n}\,{e^{cx}}

) is correct because integrating by parts and rearranging the terms leads to the given expression.

693 Easy

693 of 995

Analyzing +4 | -1

\int \frac{x^4+1}{x^6+1} d x

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because by factoring the denominator and simplifying the integrand, the integral can be expressed as the sum of two standard integrals.

694 Easy

694 of 995

Application +4 | -1

\int \sin ^3 x \cdot \cos x d x=

View answer and explanation

Correct option: B

\frac{\sin ^4 x}{4}+c

Option B (

\frac{\sin ^4 x}{4}+c

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

695 Medium

695 of 995

Application +4 | -1

\int \nolimits_0^1 \,\,{\cot ^{ - 1}}\,\,\left( {1 - x + {x^2}} \right)\;\,\,dx

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because after rewriting the integral and using integration by parts, the result is

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

696 Easy

696 of 995

Application +4 | -1

\int \log x(\log x+2) d x=

View answer and explanation

Correct option: A

x(\log x)^2+c

Option A (

x(\log x)^2+c

) is correct because the integral evaluates to this expression after substitution and simplification.

697 Medium

697 of 995

Application +4 | -1

\int \frac{3 \sin x+2 \cos x}{2 \sin x+3 \cos x} d x=l x+m \log _e|2 \sin x+3 \cos x|+c

then

View answer and explanation

Correct option: B

l=\frac{12}{13}, m=-\frac{5}{13}

Option B (

l=\frac{{12}}{{13}}, m=-\frac{{5}}{{13}}

) is correct because it is obtained by applying the formula derived from decomposing the integrand.

698 Medium

698 of 995

Application +4 | -1

\int(\sin 2 x-\cos 2 x) d x=\frac{1}{\sqrt{2}} \sin (2 x-a)+b

, then

View answer and explanation

Correct option: D

a=-\frac{5 \pi}{4}, b=

any constant

Option B ([1,3/2]) is correct because it represents the interval where the expression inside the square root is non-negative.

699 Medium

699 of 995

Application +4 | -1

\int \frac{d x}{\tan x+\cot x+\sec x+\operatorname{cosec} x}

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}(\sin x-\cos x-x)+c

Option C (

\frac{1}{2}(\sin x-\cos x-x)+c

) is correct because after simplifying the integrand and integrating, this is the resulting expression.

700 Easy

700 of 995

Application +4 | -1

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

, then

View answer and explanation

Correct option: A

A=1 / \sqrt{5}

Option A (

A=1 / \sqrt{5}

) is correct because after substitution and integration, the coefficient of the inverse tangent term is

1/\sqrt{5}

701 Medium

701 of 995

Application +4 | -1

To evaluate

\int x^3 e^{3 x^2+5} d x

, the simplest way is to

View answer and explanation

Correct option: B

Substitute

\left(3 x^2+5\right)=t

Option B (Substitute

\left(3 x^2+5\right)=t

) is correct because this substitution simplifies the integral into a manageable form.

702 Medium

702 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

\) is correct because after substituting and integrating, we obtain the given result.

703 Easy

703 of 995

Application +4 | -1

\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\sin \mathrm{x}}{1-\cos \mathrm{x}}\right) \mathrm{dx}

is equal to

View answer and explanation

Correct option: B

-\mathrm{e}^{\mathrm{x}} \cot \left(\frac{\mathrm{x}}{2}\right)+\mathrm{C}

Option B (

-e^{x} \cot \left(\frac{x}{2}\right)+C

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

704 Easy

704 of 995

Analyzing +4 | -1

\int {5^{{5^{{5^x}}}}} \cdot \,\,{5^{{5^x}}} \cdot \,\,{5^x}\;dx

is equal to

View answer and explanation

Correct option: C

\frac{{{5^{{5^{{5^x}}}}}}}{{{{\left( {\log \,\,5} \right)}^3}}}\,\, + \,\,C

Option A (

\mathrm{1}

) is correct because the limit is evaluated by simplifying the integral and applying the limit definition.

705 Easy

705 of 995

Analyzing +4 | -1

\int {\frac{{\cos \theta }}{{5 + 7\sin \theta - 2{{\cos }^2}\theta }}} \,d\theta \,\, = A\,\,{\log _e}\left| {B(\theta )} \right| + C,

where

C

is constant of integration, then

\frac{{B(\theta )}}{A}

can be

View answer and explanation

Correct option: D

\frac{{5(2\sin \theta + 1)}}{{\sin \theta + 3}}

Option D is correct because after integrating the given expression and simplifying, the result matches the form

A \log_e |B(\theta)| + C

, leading to the correct value of

\frac{{B(\theta)}}{A}

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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

706 Easy

706 of 995

Application +4 | -1

\int \frac{\left(x^3+3 x^2+3 x+1\right)}{(x+1)^5} d x=

View answer and explanation

Correct option: A

-\frac{1}{(x+1)}+c

Option A (

-\frac{{1}}{{(x+1)}}+c

) is correct because the integral simplifies to this form after simplifying the fraction.

707 Easy

707 of 995

Analyzing +4 | -1

The integral

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

is equal to

View answer and explanation

Correct option: A

- \,\,\frac{{8x + 7}}{{{x^2} + x + 1}}\,\, - \,\,\frac{{16}}{{3\,\sqrt 3 }}\,\,{\tan ^{ - 1}}\,\,\left( {\frac{{2x + 1}}{3}} \right) + C

Option A is the correct answer because the integral can be solved by breaking it into parts and using trigonometric substitution to evaluate the remaining integral.

708 Medium

708 of 995

Application +4 | -1

For what value of

a

and

b

, the equation

\int(\sin 2 x-\cos 2 x) d x=\frac{1}{\sqrt{2}} \sin (2 x-a)+b \text { holds good.}

View answer and explanation

Correct option: A

a=-\frac{5 \pi}{4}, b

is any arbitrary constant

Option A (

a = -\frac{{5 \pi}}{{4}}

, b is any arbitrary constant) is correct because integrating and comparing with the given form yields the value of a.

709 Medium

709 of 995

Analyzing +4 | -1

Let

g(x)

be a differentiable function satisfying

\frac{d}{d x}\{g(x)\}=g(x)

and

g(0)=1

, then

\int g(x)\left(\frac{2-\sin 2 x}{1-\cos 2 x}\right) d x

is equal to

View answer and explanation

Correct option: B

-g(x) \cot x+C

The correct answer is B (

-g(x) \cot x+C

) because given the properties of

g(x)

, the integral simplifies and can be solved using integration techniques.

710 Easy

710 of 995

Application +4 | -1

\int \left( {\sqrt[3]{x}} \right)\,\,\left( {\sqrt[5]{{1\,\, + \,\,\sqrt[3]{{{x^4}}}}}} \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{5}{8}\,\,{\left( {1\,\, + \,\,{x^{\frac{4}{3}}}} \right)^{\frac{6}{5}}}\,\, + \,\,c

Option C (

\frac{5}{8}\left( {1\\,\\ + \\,{x^{\frac{{4}}{{3}}}}} \right)^{\frac{{6}}{{5}}}}\\ + \\,c

) is correct because it is the result of integrating the given expression after a suitable substitution.

711 Easy

711 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

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

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

-\log(1+e^{-x})

712 Medium

712 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

\frac{\sin x-\cos x}{\sin 2 x}+c

Option B (

\frac{\sin x-\cos x}{\sin 2 x}+c

) is correct because the integral simplifies to this expression after a suitable substitution.

713 Medium

713 of 995

Application +4 | -1

\int \,\,{e^{x\log a}}\,{e^x}\;dx

is equal to

View answer and explanation

Correct option: D

\frac{{{{\left( {ae} \right)}^x}}}{{{{\log }_e}\,\,ae}}\,\, + \,\,c

Option D (

\frac{{{{\left( {ae} \right)}^x}}}{{{{\log }_e}\,ae}}\, + \,c

\) is correct because the integral of

e^{x\log a}e^x

simplifies to

\frac{{{{\left( {ae} \right)}^x}}}{{{{\log }_e}\,ae}}\, + \,c

714 Medium

714 of 995

Application +4 | -1

\int \frac{x+\sin x}{1+\cos x} \mathrm{~d} x=f(x) \tan \frac{x}{2}+C

View answer and explanation

Correct option: B

f(x)=x

Option B (

f(x)=x

) is correct because after integration and simplification, the given integral equals

x \tan \frac{x}{2}+C

715 Easy

715 of 995

Application +4 | -1

\int \,\,\frac{1}{{x + {x^5}}}\,\,dx = f\left( x \right) + c

, then the value of

\int \frac{{{x^4}}}{{x + {x^5}}}\,\;dx

equals

View answer and explanation

Correct option: A

\log \,\,x\,\, - \,\,f\,\,\left( x \right)\,\, + \,\,c

Option A (

\log \,\,x\,\, - \,\,f\,\,\left( x \right)\,\, + \,\,c

) is correct because it is obtained by manipulating the integrand and using the given integral.

716 Easy

716 of 995

Application +4 | -1

\int \frac{1+\cos 8 x}{\tan 2 x-\cot 2 x} d x=a \cos 8 x+C

, then

a=

View answer and explanation

Correct option: C

\frac{1}{16}

Option C (

\frac{1}{16}

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

\cos 8x

\frac{1}{16}

717 Medium

717 of 995

Analyzing +4 | -1

\int \frac{x^m-1}{x^{m+1} \sqrt{1-2 x^m+m x^{2 m}}} d x=\frac{f(x)}{m x^m}+c

where

f(x)

is equal to

View answer and explanation

Correct option: A

\sqrt{1-2 x^m+m x^{2 m}}

Option A (

\sqrt{{1-2 x^m+m x^{2 m}}}

) is correct because the integral simplifies to this expression after substitution and algebraic manipulation.

718 Easy

718 of 995

Application +4 | -1

\int 3^{3^{3^x}} 3^{3^x} 3^x \mathrm{~d} x

is equal to

View answer and explanation

Correct option: C

\frac{3^{3^{3^5}}}{(\log 3)^3}+C

Option C is correct because after applying the substitution

t = 3^{3^{3^x}}

, the integral simplifies to

\frac{3^{3^{3^x}}}{(\log 3)^3}+C

719 Easy

719 of 995

Application +4 | -1

The integral

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

equals

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it represents the result of the given integral after applying substitution and simplification.

720 Medium

720 of 995

Analyzing +4 | -1

The value of the integral

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

View answer and explanation

Correct option: C

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

Option C is correct because the integral can be solved by rewriting the integrand and using integration by parts, leading to the result e^x / (1 + x²) + c.

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