If $\int \sin 5 x \cos 3 x d x=-\frac{\cos 8 x}{16}+A$, then $A=$

$\frac{\sin 2 x}{16}+$ constant

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

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

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

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

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

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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 91-120 of 995 questions

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91 Easy

91 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{1}{3}(1+\log x)^3+c

Option C is correct because the integral simplifies to

\frac{1}{3}(1+\log x)^3+c

after a simple substitution.

92 Easy

92 of 995

Understanding +4 | -1

\int \frac{f(x) g^{\prime}(x)-g(x) f^{\prime}(x)}{f(x) \cdot g(x)}\{\log g(x)-\log f(x)\} d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}\left\{\log _e \frac{g(x)}{f(x)}\right\}^2+C

Option B is correct because the integral simplifies to a form where the substitution

u = \log_e \frac{{g(x)}}{{f(x)}}

can be used, resulting in

\frac{1}{2}u^2 + C

93 Medium

93 of 995

Application +4 | -1

The value of

\int \frac{d x}{3-2 x-x^2}

will be

View answer and explanation

Correct option: A

\frac{1}{4} \log \left(\frac{3+x}{1-x}\right)+c

Option A (

\frac{1}{4} \log \left(\frac{3+x}{1-x}\right)+c

) is correct because it is the result of integrating the given function after completing the square and using partial fractions.

94 Medium

94 of 995

Application +4 | -1

f(0)=f^{\prime}(0)=0

and

f^{\prime \prime}(x)=\tan ^2 x

then

f(x)

View answer and explanation

Correct option: A

\log \sec x-\frac{1}{2} x^2

Option A (

\log \sec x-\frac{1}{2} x^2

) is correct because integrating

f''(x) = \tan^2 x

twice and applying the initial conditions

f(0) = f'(0) = 0

yields this result.

95 Medium

95 of 995

Application +4 | -1

\int \frac{x^2}{(a+b x)^2} d x

equals to:

View answer and explanation

Correct option: B

\frac{1}{b^3}\left(a+b x-2 a \ln |a+b x|-\frac{a^2}{a+b x}\right)+C

Option B (

\frac{1}{b^3}\left(a+b x-2 a \ln |a+b x|-\frac{a^2}{a+b x}\right)+C

) is correct because it is the result of correctly integrating the given expression using substitution and partial fractions.

96 Medium

96 of 995

Analyzing +4 | -1

Let

\int \,\,\frac{{{x^2}}}{{\sqrt {1 - x} }}\,\,dx = p\,\,\sqrt {\left( {1 - x} \right)} \,\,\left( {3{x^2} + 4x + 8} \right)

, then value of

p

View answer and explanation

Correct option: A

\frac{{ - \,2}}{{15}}

Option A (

\frac{{ - \,2}}{{15}}

) is correct because substituting

1-x = t^2

and integrating yields the correct value for p.

97 Medium

97 of 995

Application +4 | -1

\int \frac{d x}{(x+1)^{2 / 3}(x-1)^{4 / 3}}

equals

View answer and explanation

Correct option: A

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

Option A is correct because the integral is solved by substitution, where (x+1)/(x-1) = t³, simplifying the integral to a power rule.

98 Easy

98 of 995

Application +4 | -1

Let

f\left( x \right)

be a function such that,

f\left( 0 \right) = f'\left( 0 \right) = 0,\,\,f''\left( x \right) = {\sec ^4}\,\,x\,\, + \,\,4

, then the function is

View answer and explanation

Correct option: B

\frac{2}{3}\,\,\log \,\,|(\sec \,\,x)|\,\, + \,\,\frac{1}{6}\,\,{\tan ^2}x + 2{x^2}

Option B (

\frac{2}{3}\,\,\log \,|(\sec \,x)\,|\,\,+\,\,\frac{1}{6}\,\,{\tan ^2}x + 2{x^2}

) is correct because it is the result of integrating the given second derivative and applying the initial conditions.

99 Easy

99 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A is correct because the integral is solved using partial fraction decomposition, resulting in the given expression.

100 Medium

100 of 995

Analyzing +4 | -1

\int \,\,\frac{{{{\sin }^8}\,\,x\,\, - \,\,{{\cos }^8}\,\,x}}{{1 - 2\,\,{{\sin }^2}\,\,x\;\,\,{{\cos }^2}\,\,x}}\,\,\;dx = A\,\,\sin \,\,\,2x + B

, then

View answer and explanation

Correct option: A

A = - \,\,\frac{1}{2}

Option A is correct because after simplifying the integral, the coefficient of

\sin \,2x

- \,\frac{1}{2}

101 Easy

101 of 995

Application +4 | -1

\int \cos x \sqrt{4-\sin ^2 x} d x=

View answer and explanation

Correct option: B

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

Option B is correct because the integral can be solved using trigonometric substitution, resulting in

\frac{1}{2} \sin x \sqrt{4-\sin ^2 x}+2 \sin ^{-1}(\frac{1}{2} \sin x)+c

102 Medium

102 of 995

Application +4 | -1

\int \limits_0^2 \left( {x - {{\log }_2}\,\,a} \right)dx = 2\,\,{\log _2}\,\,\left( {\frac{2}{a}} \right)

, if

View answer and explanation

Correct option: A

a = 2

Option A (

a = 2

) is correct because substituting

a=2

into the integral equation satisfies the given condition.

103 Medium

103 of 995

Application +4 | -1

\int {\sin ^{ - 1}}\,\,\left( {\frac{{2x}}{{1 + {x^2}}}} \right)\,\,dx = f\left( x \right)\,\, - \,\,\log \,\,\left( {1 + {x^2}} \right)\,\, + \,\,c

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: A

2x\,\,{\tan ^{ - 1}}\,\,x

Option A (

2x\,{\tan ^{ - 1}}\,x

) is correct because it is derived by using trigonometric substitution and integration by parts.

104 Easy

104 of 995

Analyzing +4 | -1

\int \,\,x\,\mathrm{ }f\left( x \right)\,\,dx = \,\,\frac{{f\left( x \right)}}{2}

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: D

\frac{{{e^{{x^2}}}}}{2}

Option D (

\frac{{{e^{{x^2}}}}}{2}

) is correct because when substituted into the integral equation, it satisfies the given condition.

105 Easy

105 of 995

Application +4 | -1

\int \sin 5 x \cos 3 x d x=-\frac{\cos 8 x}{16}+A

, then

A=

View answer and explanation

Correct option: B

-\frac{\cos 2 x}{4}+

constant

Option B,

-\frac{\cos 2 x}{4}+

constant, is the correct answer because it is derived from integrating the product of sine and cosine functions using trigonometric identities.

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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

106 Easy

106 of 995

Application +4 | -1

\int \frac{d x}{\sin ^2 x+\tan ^2 x}

View answer and explanation

Correct option: A

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

Option A (

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

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

107 Medium

107 of 995

Application +4 | -1

\int \frac{x+\sin x}{1+\cos x} d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

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

108 Easy

108 of 995

Application +4 | -1

\int x \cos ^2 x d x=

View answer and explanation

Correct option: B

\frac{x^2}{4}+\frac{1}{4} x \sin 2 x+\frac{1}{8} \cos 2 x+c

Option B (

\frac{x^2}{4}+\frac{1}{4} x \sin 2 x+\frac{1}{8} \cos 2 x+c

) is correct because it is the result of integrating

x \cos ^2 x

using trigonometric identities and integration by parts.

109 Easy

109 of 995

Analyzing +4 | -1

The value of

\int \frac{{x\,\,\cos \,\,x + 1}}{{\sqrt {2{x^3}\,{e^{\sin x}} + \,\,{x^2}} }}\,\,dx

View answer and explanation

Correct option: A

ln\;\left| {\,\frac{{\sqrt {2x\,\;{e^{\sin x}} + \,\,1} - 1}}{{\sqrt {2x\,\;{e^{\sin x}} + \,\,1} + 1}}\,} \right| + C

Option A (

ln\;\left| {\\,\frac{{\sqrt {2x\\;{{e^{\sin x}} + \\,1} - 1}}{{\sqrt {2x\\;{{e^{\sin x}} + \\,1} + 1}}\} \right| + C

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

110 Medium

110 of 995

Application +4 | -1

The value of

\int \frac{d x}{\sqrt{(x-a)(b-x)}}

, is

View answer and explanation

Correct option: A

2 \sin ^{-1} \sqrt{\frac{x-a}{b-a}}+C

Option A (

2 \sin ^{-1} \sqrt{\frac{{x-a}}{{b-a}}}+C

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

111 Easy

111 of 995

Application +4 | -1

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

View answer and explanation

Correct option: A

\frac{1}{x} 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{1}{x}

and

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

112 Easy

112 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the integral is solved using substitution and partial fraction decomposition.

113 Easy

113 of 995

Application +4 | -1

The value of

\int \frac{\sin x-\cos x}{\sqrt{\sin 2 x-1 / 2}} \mathrm{~d} x

View answer and explanation

Correct option: B

-\log |(\sin x+\cos x)+\sqrt{\sin 2 x-1 / 2}|+C

Option B (

-\log |(\sin x+\cos x)+\sqrt{{\sin 2 x-1 / 2}}|+C

) is correct because the integral can be solved by substitution, where

t = \sin x + \cos x

114 Medium

114 of 995

Analyzing +4 | -1

\int \frac{d x}{\left(x^2-4\right) \sqrt{x+1}}=\frac{1}{k} \log _e\left|\frac{\sqrt{x+1}-\sqrt{3}}{\sqrt{x+1}+\sqrt{3}}\right|

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

then

k

equals

View answer and explanation

Correct option: B

4 \sqrt{3}

Option B (

4 \sqrt{3}

) is correct because after applying a trigonometric substitution and simplifying, the constant

k

is found to be

4 \sqrt{3}

115 Easy

115 of 995

Application +4 | -1

\int \frac{x+3}{(x+4)^2} e^x d x

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the integral can be solved by rewriting the integrand and applying the formula

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

116 Easy

116 of 995

Application +4 | -1

Let

f(x)=\frac{\sqrt{\tan x}}{\sin x \cos x}

and

F(x)

is its antiderivative, if

F(\pi / 4)=6

then

F(x)

is equal to

View answer and explanation

Correct option: C

2(\sqrt{\tan x}+2)

Option C (

2(\sqrt{{1+\tan x}}+2)

) is correct because it represents the antiderivative of the given function with the correct constant of integration.

117 Easy

117 of 995

Analyzing +4 | -1

The value of

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

=-\frac{1}{2}\left\{\log \left(z+\frac{3}{2}+\sqrt{z^2+3 z+3}\right)\right\}+\frac{1}{2}\left|\log s-\frac{1}{2}+\sqrt{s^2-s+1}\right|+C

and

s-z=\frac{k}{x}

, then value of

k

, is

View answer and explanation

Correct option: B

Option B (2) is correct because after solving the integral and comparing with the given expression, the value of k is found to be 2.

118 Easy

118 of 995

Application +4 | -1

\int e^{\sqrt{x}} d x

is equal to

View answer and explanation

Correct option: C

2(\sqrt{x}-1) e^{\sqrt{x}}+A

Option C (

2(\sqrt{x}-1) e^{\sqrt{x}}+A

) is correct because substituting

\sqrt{x} = t

simplifies the integral, and integration by parts yields the correct result.

119 Easy

119 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because after performing polynomial long division and partial fraction decomposition, the integral evaluates to

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

120 Easy

120 of 995

Application +4 | -1

\int \frac{1}{\sqrt{x}} \sin \sqrt{x} d x=

View answer and explanation

Correct option: B

-2 \cos \sqrt{x}+c

Option B (

-2 \cos \sqrt{x}+c

) is correct because the integral evaluates to this after a simple substitution.

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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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