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

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

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

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

$\smallint \,\,\frac{{({{\sec }^2}\,\,x\,\, - \,\,7)}}{{{{\sin }^7}\,\,x}}\,\,\;dx$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 $f(x)=$

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

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

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

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

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

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

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

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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 1-30 of 411 questions

1 Medium

1 of 411

Application +4 | -1

f(x)

is differentibale and

\int\limits_{0}^{t^2} x f(x) d x=\frac{2}{5} t^5

then

f\left(\frac{4}{25}\right)

equals

View answer and explanation

Correct option: A

2 / 5

Option A (2 / 5) is correct because it is the value of f(4/25) obtained after differentiating both sides of the given equation and substituting t=2/5.

2 Medium

2 of 411

Application +4 | -1

\int {e}^{{x}^2}\left(2-\frac{1}{{x}^2}\right) {de}={e}^{{x}^2} {f}({x})+{C}

and

{f}\left(\frac{1}{2}\right)=2

, then

f(1)

is equal to (where

{C}

is an arbitrary constant)

View answer and explanation

Correct option: A

Option A (1) is correct because after integrating and comparing with the given form, f(x) is found to be 1/x, and f(1) equals 1.

3 Medium

3 of 411

Application +4 | -1

The value of

\int {\frac{{1 + x + {x^2}}}{{1 + {x^2}}}{e^{{{\tan }^{ - 1}}x}}dx}

is equal to

View answer and explanation

Correct option: C

x{e^{{{\tan }^{ - 1}}x}} + c

Option C (

x{e^{{{\tan }^{ - 1}}x}} + c

) is correct because after substituting

t = \tan^{-1}x

and integrating by parts, the integral evaluates to this expression.

4 Medium

4 of 411

Application +4 | -1

The value of

\int \limits_0^{\pi /2} \,\,\frac{{{{\sin }^3}\,\,x\,\,\cos \,\,x}}{{{{\sin }^4}\,\,x\,\, + \,\,{{\cos }^4}\,\,x}}\,\,dx

, is

View answer and explanation

Correct option: A

\pi /8

Option A (π/8) is correct because by using the property of definite integrals and trigonometric identities, the integral simplifies to π/8.

5 Medium

5 of 411

Analyzing +4 | -1

y(x-y)^2=x

, then

\int \frac{d x}{(x-3 y)}=\frac{m}{n} \ln \left[(x-y)^2-1\right]

. Then

(m+2 n)

is equal to

View answer and explanation

Correct option: C

Option C is correct because after solving the differential equation and substituting back, the value of (m+2n) is found to be 5.

6 Medium

6 of 411

Application +4 | -1

\mathrm{I}=\int_{-2}^0\left[\mathrm{x}^3+3 \mathrm{x}^2+3 \mathrm{x}+(\mathrm{x}+1) \cos (\mathrm{x}+1)\right] \mathrm{dx}

, then

\mathrm{I}

equals:

View answer and explanation

Correct option: C

-2

Option A is incorrect. The correct answer should be

c^3

but the provided solution and explanation are flawed and do not lead to the correct conclusion. The problem involves arithmetic progressions and their sums.

7 Medium

7 of 411

Analyzing +4 | -1

\int \frac{\cos ^4 x}{\sin ^2 x} d x=A \cot x+B \sin 2 x+\frac{C}{2} x+D

, then

View answer and explanation

Correct option: B

B=-1 / 4, C=-3

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

B = -\frac{1}{4}

and

C = -3

8 Medium

8 of 411

Application +4 | -1

\int \sin 51 x(\sin x)^{49} d x=

View answer and explanation

Correct option: A

\frac{\sin 50 x(\sin x)^{50}}{50}+c

Option A (

\frac{{\sin 50 x(\sin x)^{{50}}}}{{50}}+c

) is correct because the integral simplifies to this form after applying trigonometric identities and integration by parts.

9 Medium

9 of 411

Analyzing +4 | -1

The value of

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

is equal to

View answer and explanation

Correct option: A

\log \left|\frac{1+\tan x}{\tan x}\right|-\cot x+C

Option A (

\log \left|\frac{{1+\tan x}}{{\tan x}}\right|-\cot x+C

) is correct because the integral can be solved by dividing the numerator and denominator by cos³(x) and using partial fractions after substituting tan(x) = t.

10 Medium

10 of 411

Application +4 | -1

\int e^{-2 x} \sin 3 x d x=

View answer and explanation

Correct option: D

-\frac{1}{13} e^{-2 x}[2 \sin 3 x+3 \cos 3 x]+c

Option D (

-\frac{{1}}{{13}} e^{-2 x}[2 \sin 3 x+3 \cos 3 x]+c

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

11 Medium

11 of 411

Application +4 | -1

The value of

\int \frac{d x}{\sin x \cdot \sin (x+\alpha)}

is equal to

View answer and explanation

Correct option: A

\operatorname{cosec} \alpha \,\ell n \left|\frac{\sin x}{\sin (x+\alpha)}\right|+C

Option A (

\operatorname{cosec} \alpha \,\ell n \left|\frac{{\sin x}}{{\sin (x+\alpha)}}\right|+C

) is correct because it results from using trigonometric identities and properties of logarithms.

12 Medium

12 of 411

Application +4 | -1

\int \frac{d x}{\sqrt{x}(1+\sqrt[4]{x})^{10}}=a(1+\sqrt[4]{x})^{-8}+b(1+\sqrt[4]{x})^{-9}

then

a+b

View answer and explanation

Correct option: B

-\frac{1}{18}

Option B (

-\frac{1}{18}

) is correct because after evaluating the integral and comparing coefficients,

a+b

equals this value.

13 Medium

13 of 411

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

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

Option C (

\log \,|\,(1 + x{e^x})\,| + c

) is correct because by using substitution, the integral simplifies to this logarithmic form.

14 Medium

14 of 411

Application +4 | -1

\int \sin 2 x \cos 3 x d x=

View answer and explanation

Correct option: B

\frac{1}{2}\left(\cos x-\frac{1}{5} \cos 5 x\right)+c

Option B is correct because the integral of

\sin 2x \cos 3x

is solved using the product-to-sum trigonometric identities.

15 Medium

15 of 411

Application +4 | -1

\int \frac{x}{x^4-1} d x=

View answer and explanation

Correct option: A

\frac{1}{4} \log \left[\frac{x^2-1}{x^2+1}\right]+c

Option A is correct because the integral is solved by substituting

t = x^2

and using partial fraction decomposition.

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

16 Medium

16 of 411

Application +4 | -1

3a\,\,\int \nolimits_0^1 \,\,{\left( {\frac{{ax - 1}}{{a - 1}}} \right)^2}dx

is equal to

View answer and explanation

Correct option: A

a - 1 + {\left( {a - 1} \right)^{ - 2}}

Option A is correct because the integral evaluates to the given expression after simplification, matching the form in option A.

17 Medium

17 of 411

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.

18 Medium

18 of 411

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.

19 Medium

19 of 411

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.

20 Medium

20 of 411

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.

21 Medium

21 of 411

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.

22 Medium

22 of 411

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.

23 Medium

23 of 411

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A is correct because the integral can be solved by using a suitable trigonometric substitution.

24 Medium

24 of 411

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.

25 Medium

25 of 411

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.

26 Medium

26 of 411

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.

27 Medium

27 of 411

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.

28 Medium

28 of 411

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

29 Medium

29 of 411

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

30 Medium

30 of 411

Application +4 | -1

\int \sqrt{\frac{x^6}{a^8+x^8}} d x=

View answer and explanation

Correct option: B

\frac{1}{4} \ln \left|x^4+\sqrt{a^8+x^8}\right|+c

Option B (

\frac{{1}}{{4}} \ln |x^4+\sqrt{{a^8+x^8}}|+c

) is correct because the integral is solved using substitution and the standard integral formula for

\int \frac{{dx}}{{\sqrt{{x^2+a^2}}}}

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 411 single-correct-answer MCQs on INDEFINITE INTEGRATION.
✓ Try each question before opening its answer. Read the explanation where one is available.
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