$\smallint \,\,\frac{{({{\sec }^2}\,\,x\,\, - \,\,7)}}{{{{\sin }^7}\,\,x}}\,\,\;dx$is equal to

$\frac{{\cos \,\,x}}{{{{\sin }^7}\,\,x}}\,\, + \,\,c$

$\frac{{\sin \,\,x}}{{{{\cos }^7}\,\,x}}\,\, + \,\,c$

$\frac{{\sin \,\,x}}{{{{\tan }^7}\,\,x}}\,\, + \,\,c$

$\smallint \,\,\frac{{dx}}{{x\left( {{x^7} + 1} \right)}}$ is equal to

$\log \,\,\left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$

$\frac{1}{7}\,\,\log \,\,\left( {\frac{{{x^7} + 1}}{{{x^7}}}} \right) + c$

The value of $\smallint \frac{{dx}}{{x\,\, + \,\,\sqrt {x\,\, - \,\,1} }}$ is

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

$\log \,\,\left( {x\,\, + \,\,\sqrt {x - 1} } \right)\,\, + \,\,c$

If $\int \frac{e^x-1}{e^x+1} d x=f(x)+C$, then $f(x)=$

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

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

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

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

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

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

If $\smallint \,\,\frac{{4{e^x} + \,\,6{e^{ - x}}}}{{9{e^x} - \,\,4{e^{ - x}}}}\,\,dx = Ax + B\,\,\log \,\,\left( {9{e^{2x}} - \,\,4} \right) + C$, then

$A = - \,\,\frac{3}{2},\,\,\;B = \frac{{35}}{{36}},\;\,\,C = 0$

$A = \frac{{35}}{{36}},\;\,\,B = - \,\,\frac{3}{2},\;\,\,C \in R$

$\smallint \,\,{\tan ^{ - 1}}\,\,x\,\,dx = ...\,\, + c$

$\frac{1}{{1\,\, + \,\,{x^2}}}$

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

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

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

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

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

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

$\smallint \,\,\sqrt[3]{x}\,\,\sqrt[7]{{1\,\, + \,\,\sqrt[3]{{{x^4}}}\,}}dx$ is equal to

$\frac{{32}}{{21}}\,\,{\left\{ {1\,\, + \,\,\sqrt[3]{{{x^4}}}} \right\}^{8/7}} + \,\,C$

$\frac{7}{{32}}\,\,{\left\{ {1\,\, + \,\,\sqrt[3]{{{x^4}}}} \right\}^{8/7}} + \,\,C$

Back to INDEFINITE INTEGRATION

Your Progress

0 / 995 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 61-90 of 995 questions

Preparing practice and saved progress…

61 Easy

61 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

(2 x-4 \sqrt{x}+4) e^{\sqrt{x}}+c

Option B (

(2 x-4 \sqrt{x}+4) e^{\sqrt{x}}+c

) is correct because it is the result of integrating

\sqrt{x} e^{\sqrt{x}}

using substitution and integration by parts twice.

62 Medium

62 of 995

Application +4 | -1

\int \,\,\frac{{({{\sec }^2}\,\,x\,\, - \,\,7)}}{{{{\sin }^7}\,\,x}}\,\,\;dx

is equal to

View answer and explanation

Correct option: A

\frac{{\tan \,\,x}}{{{{\sin }^7}\,\,x}}\,\, + \,\,c

Option A is correct because the given integral simplifies to the derivative of (tan x) / (sin⁷ x), making the solution (tan x) / (sin⁷ x) + c.

63 Easy

63 of 995

Application +4 | -1

Let

f(x)=1+3^x \log 3

and

F(x)

be its antiderivative which assume the value 7 for

x=2

. The value of

x

for which the curve

F(x)

cuts the abscissa axis is

View answer and explanation

Correct option: B

Option B (1) is correct because it is the value of x that makes F(x) = 0, after finding the constant of integration.

64 Easy

64 of 995

Application +4 | -1

The solution of the equation

\int \limits_{\log \,\,2}^x \,\,\frac{1}{{{e^x} - 1}}\,\,dx = \log \,\,\frac{3}{2}

is given by

x =

View answer and explanation

Correct option: C

\log \,\,4

Option C (

\log \,4

) is correct because solving the definite integral and equating it to

\log \,\frac{3}{2}

leads to

x = \log 4

65 Medium

65 of 995

Application +4 | -1

\int \,\,\frac{{dx}}{{x\left( {{x^7} + 1} \right)}}

is equal to

View answer and explanation

Correct option: B

\frac{1}{7}\,\,\log \,\,\left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c

Option B (

\frac{1}{7}\,\log \,\left( {\frac{{{x^7}}}{{{x^7} + 1}}} \right) + c

) is correct because after substitution and partial fraction decomposition, the integral evaluates to this expression.

66 Easy

66 of 995

Application +4 | -1

Let

\alpha \in \left( {0,\,\,\pi /2} \right)

be fixed. If the integral

\int {\frac{{\tan x + \tan \alpha }}{{\tan x - \tan \alpha }}dx = A\left( x \right)\cos 2\alpha + B\left( x \right)}

\sin 2\alpha + C\,

, where

C

is a constant of integration, then the functions

A\left( x \right)

and

B\left( x \right)

are respectively

View answer and explanation

Correct option: B

x - \alpha

and

{\log _e}\left| {\sin \left( {x - \alpha } \right)} \right|

Option B (

x - \alpha

and

{\log _e}\left| {\sin \left( {x - \alpha } \right)} \right|

) is correct because these are the functions A(x) and B(x) obtained after evaluating the given integral.

67 Medium

67 of 995

Analyzing +4 | -1

The value of

\int \frac{{dx}}{{x\,\, + \,\,\sqrt {x\,\, - \,\,1} }}

View answer and explanation

Correct option: C

\log \,\,\left( {x\,\, + \,\,\sqrt {x - 1} } \right) - \,\,\frac{2}{{\sqrt 3 }}\,\,{\tan ^{ - 1}}\,\,\left( {\frac{{2\,\sqrt {x\,\, - \,\,1} + \,\,1}}{{\sqrt 3 }}} \right)\,\, + \,\,c

Option C (

\log \,\,\left( {x\, + \,\sqrt {x - 1} } \right) - \,\,\frac{2}{{\sqrt 3 }}\,{\tan ^{ - 1}}\,\,\left( {\frac{{2\,\sqrt {x\, - \,1} + \,1}}{{\sqrt 3 }}} \right)\, + \,c

) is correct because after substitution and integration, the integral evaluates to the given expression.

68 Easy

68 of 995

Application +4 | -1

\int x \sin ^2 x d x=

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the result of integrating

x \sin^2 x

using trigonometric identities and integration by parts.

69 Easy

69 of 995

Application +4 | -1

A value of

\alpha

such that

\int\limits_\alpha ^{\alpha + 1} {\frac{{dx}}{{\left( {x + a} \right)\left( {x + \alpha + 1} \right)}} = {{\log }_e}\left( {\frac{9}{8}} \right)}

View answer and explanation

Correct option: A

- 2

Option A (-2) is correct because it is a value of

\alpha

that satisfies the given integral equation after solving the integral and substituting the limits.

70 Easy

70 of 995

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.

71 Medium

71 of 995

Analyzing +4 | -1

\int 7^{7^{7^x}} \cdot 7^{7^x} \cdot 7^x d x

is equal to

View answer and explanation

Correct option: A

\frac{7^{7^{7^x}}}{\left(\log _e 7\right)^3}+C

Option A is correct because the integral can be solved by recognizing the pattern and using substitution, leading to the given result.

72 Easy

72 of 995

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.

73 Medium

73 of 995

Application +4 | -1

\int \frac{e^x-1}{e^x+1} d x=f(x)+C

, then

f(x)=

View answer and explanation

Correct option: C

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

Option C is correct because the integral simplifies to 2log(e^x + 1) - x + C after algebraic manipulation and integration.

74 Easy

74 of 995

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.

75 Medium

75 of 995

Analyzing +4 | -1

f2(x) = {\tan ^{ - 1}}(\sec x + \tan x), - \frac{\pi }{2} < x < \frac{\pi }{2},\,\,{\text{and}}\,\,f(0) = 0,\,\,{\text{then}}\,\,f(1)

is equal to:

View answer and explanation

Correct option: A

\frac{\pi+1}{4}

Option A (

\frac{{\pi+1}}{{4}}

) is correct because it is the value of f(1) obtained after integrating f'(x) and applying the initial condition f(0) = 0.

Explore Other Chapters in MATHEMATICS

View All →

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 995

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.

77 Medium

77 of 995

Analyzing +4 | -1

\int \,\,\frac{{4{e^x} + \,\,6{e^{ - x}}}}{{9{e^x} - \,\,4{e^{ - x}}}}\,\,dx = Ax + B\,\,\log \,\,\left( {9{e^{2x}} - \,\,4} \right) + C

, then

View answer and explanation

Correct option: C

A = - \,\,\frac{3}{2},\;\,\,B = \frac{{35}}{{36}},\;\,\,C \in R

Option C (

A = - \,\frac{3}{2},\,\;B = \frac{{35}}{{36}},\,\;C \in R

) is correct because after integrating the given function and comparing with the given form, we find the values of A and B.

78 Easy

78 of 995

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

79 Medium

79 of 995

Application +4 | -1

\int \,\,{\tan ^{ - 1}}\,\,x\,\,dx = ...\,\, + c

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the result of integrating

\tan^{-1}x

using integration by parts.

80 Easy

80 of 995

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.

81 Medium

81 of 995

Application +4 | -1

\int(\sin x)^{-3 / 2}(\cos x)^{-5 / 2} d x=

View answer and explanation

Correct option: C

-2 \sqrt{\cot x}+\frac{2}{3} \sqrt{\tan ^3 x}+c

Option C (

-2 \sqrt{{\cot x}}+\frac{{2}}{{3}} \sqrt{{\tan ^3 x}}+c

) is correct because it is the result of integrating the given expression after trigonometric manipulation and substitution.

82 Medium

82 of 995

Application +4 | -1

\int \cos \left(\ln \frac{x}{a}\right) d x

is equal to

View answer and explanation

Correct option: B

\frac{x}{2}\left(\cos \left(\ln \frac{x}{a}\right)+\sin \left(\ln \frac{x}{a}\right)\right)+c

Option B (

\frac{x}{2}(\cos(\ln \frac{x}{a})+\sin(\ln \frac{x}{a}))) + c

) is correct because it is the result of integrating

\cos(\ln \frac{x}{a})

using integration by parts.

83 Easy

83 of 995

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.

84 Medium

84 of 995

Application +4 | -1

\int\left(x^x\right)^x\left(2 x \log _e x+x\right) d x

is equal to

View answer and explanation

Correct option: B

\left(x^x\right)^x+C

Option B

((x^x)^x+C)

is correct because the integral simplifies to the derivative of

x^{x^2}

, making the solution

(x^x)^x+C

85 Medium

85 of 995

Analyzing +4 | -1

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

, then

f(x)

View answer and explanation

Correct option: D

\cot x

Option D (

\cot x

) is correct because by rewriting the integrand and integrating,

f(x)

is found to be

\cot x

86 Easy

86 of 995

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.

87 Easy

87 of 995

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}

88 Easy

88 of 995

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.

89 Medium

89 of 995

Application +4 | -1

\int \,\,\sqrt[3]{x}\,\,\sqrt[7]{{1\,\, + \,\,\sqrt[3]{{{x^4}}}\,}}dx

is equal to

View answer and explanation

Correct option: A

\frac{{21}}{{32}}\,\,{\left\{ {1\,\, + \,\,\sqrt[3]{{{x^4}}}} \right\}^{8/7}} + \,\,C

Option A is correct because the integral is solved using substitution, resulting in the given expression.

90 Medium

90 of 995

Application +4 | -1

The integral of

f(x)=3^x \cos x

is given by

View answer and explanation

Correct option: C

\frac{3^x(\sin x+\cos \log 3)}{\left(1+(\log 3)^2\right)}

Option D is correct because the integral of

3^x \cos x

\frac{3^x(\log 3 \sin x+\cos x)}{\left(1+(\log 3)^2\right)}+C

after integrating by parts twice.

Revision notes and key takeaways

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

Showing 61-90 of 995 questions

Your Progress

Indefinite integration practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice