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

$\frac{1}{2}\,\,\log \,\,\left| {\,1 + \tan \,\,\frac{x}{2}\,} \right| + c$

$2\,\,\log \,\,\left| {\,1 + \tan \,\,\frac{x}{2}\,} \right| + c$

$\frac{1}{2}\,\,\log \,\,\left| {\,1 - \tan \,\,\frac{x}{2}\,} \right| + c$

$\smallint \,\,\frac{{\left( {\sin \,\,{\rm{\theta }}\,\, + \,\,\cos \,\,{\rm{\theta }}} \right)}}{{\sqrt {\sin \,\,2{\rm{\theta }}} }}\,\,d{\rm{\theta }}$ is equal to

${\sin ^{ - 1}}\,\,\left( {\sin \,\,{\rm{\theta }} - \cos \,\,{\rm{\theta }}} \right)\,\, + \,\,c$

$\log \,\,\left| {\,\cos \,\,{\rm{\theta }}\,\, - \,\,\sin \,\,{\rm{\theta }}\,\, + \,\,\sqrt {\sin \,\,2{\rm{\theta }}} \,} \right| + c$

$\log \,\,\left| {\,\sin \,\,{\rm{\theta }}\,\, - \,\,\cos \,\,{\rm{\theta }}\,\, + \,\,\sqrt {\sin \,\,2{\rm{\theta }}\,} } \right| + c$

${\sin ^{ - 1}}\,\,\left( {\sin \,\,{\rm{\theta }}\,\, + \,\,\cos \,\,{\rm{\theta }}} \right)\,\, + \,\,c$

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 · 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 61-90 of 746 questions

61 Easy

61 of 746

Application +4 | -1

The value of

\int x \log x(\log x-1) d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral simplifies to a form that can be directly integrated using substitution.

62 Easy

62 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral simplifies to this expression after integration by parts.

63 Easy

63 of 746

Application +4 | -1

The function

f

whose graph passes through

(\pi / 4,0)

and whose derivative is

\frac{\log (\tan x)}{\sin x \cos x}

is given by

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because integrating the derivative and using the given point yields this function.

64 Easy

64 of 746

Application +4 | -1

\int \frac{1}{{1\,\, + \,\,\cos \,\,x\,\, + \,\,\sin \,\,x}}\;dx

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the result of substituting

t = \tan \frac{x}{2}

and integrating the simplified expression.

65 Easy

65 of 746

Analyzing +4 | -1

Let

f(x)

be a polynomial of degree three satisfying

f(0)=-1

and

f(1)=0

. Also, 0 is a stationary point of

f(x)

. If

f(x)

does not have an extremum at

x=0

, then the value of the integral

\int \frac{f(x)}{x^3-1} d x

, is

View answer and explanation

Correct option: B

x+C

Option B (

x+C

) is correct because after substituting the given conditions and simplifying, the integral evaluates to

x+C

66 Easy

66 of 746

Analyzing +4 | -1

Let

I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x.

Then, for an arbitrary constant

c

, the value of

J-I

equals to

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because after substituting and simplifying the integrals, the difference J-I results in this logarithmic expression.

67 Easy

67 of 746

Application +4 | -1

\int \sec ^4 x \tan x d x=

View answer and explanation

Correct option: A

\frac{1}{4} \sec ^4 x+c

Option A (

\frac{1}{4} \sec ^4 x+c

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

68 Easy

68 of 746

Application +4 | -1

Evaluate

\left[\int\limits_{0}^\pi \frac{d x}{1+2 \sin ^2 x}\right]

where

[\cdot]

denotes greatest integer function.

View answer and explanation

Correct option: A

Option A (1) is correct because the definite integral evaluates to

\frac{{\pi}}{{\sqrt{{3}}}}

, and the greatest integer function of that value is 1.

69 Easy

69 of 746

Application +4 | -1

\int \,\,\frac{{\left( {\sin \,\,\mathrm{\theta }\,\, + \,\,\cos \,\,\mathrm{\theta }} \right)}}{{\sqrt {\sin \,\,2\mathrm{\theta }} }}\,\,d\mathrm{\theta }

is equal to

View answer and explanation

Correct option: C

{\sin ^{ - 1}}\,\,\left( {\sin \,\,\mathrm{\theta } - \cos \,\,\mathrm{\theta }} \right)\,\, + \,\,c

Option C (

{\sin ^{ - 1}}\,\,\left( {\sin \,\,\mathrm{\theta } - \cos \,\,\mathrm{\theta }} \right)\,\, + \,\,c

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

70 Easy

70 of 746

Understanding +4 | -1

\int e^{a x} \cos b x d x

is equal to

View answer and explanation

Correct option: A

Real part of

\int e^{(a+b i) x} d x

Option A (Real part of

\int e^{(a+b i) x} d x

) is correct because Euler's formula allows us to express

\cos bx

as the real part of

e^{ibx}

71 Easy

71 of 746

Analyzing +4 | -1

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

, then

View answer and explanation

Correct option: B

f\,\left( x \right) = \log \,\,x,\,\,A = - \,2

Option B (f(x) = log x, A = -2) is correct because integrating by parts twice and comparing with the given form reveals these values for f(x) and A.

72 Easy

72 of 746

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.

73 Easy

73 of 746

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

74 Easy

74 of 746

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.

75 Easy

75 of 746

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.

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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

76 Easy

76 of 746

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

77 Easy

77 of 746

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.

78 Easy

78 of 746

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.

79 Easy

79 of 746

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.

80 Easy

80 of 746

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.

81 Easy

81 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

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

Option D is correct because the integral simplifies to a standard form after using trigonometric identities and a suitable substitution.

82 Easy

82 of 746

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.

83 Easy

83 of 746

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}

84 Easy

84 of 746

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.

85 Easy

85 of 746

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

86 Easy

86 of 746

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

87 Easy

87 of 746

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.

88 Easy

88 of 746

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.

89 Easy

89 of 746

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.

90 Easy

90 of 746

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

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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