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

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

$\smallint \sqrt {1\,\, + \,\,\cos \,\,x} \,dx$ is equal to

$2\,\,\sqrt 2 \,\,\cos \,\,\frac{x}{2}\,\, + \,\,c$

$\sqrt 2 \,\,\cos \,\,\frac{x}{2}\,\, + \,\,c$

$\smallint \,\,\left[ {{e^{a\,\,\log \,\,x}}\,\, + \,\,{e^{x\,\,\log \,\,a}}} \right]\,\,dx\,\, = ... + \,\,c\,\,;\,\,a,\,\,x\,\, > 1$

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

$\frac{{{x^{a - 1}}}}{{a - 1}}\,\, + \,\,{a^x}\,\,.\,\,\log \,\,a$

$\frac{{{e^{a\,\,\log \,\,x}}}}{{a/x}}\,\, + \,\,\frac{{{e^{x\,\,\log \,\,a}}}}{x}$

If ${I_1} = \smallint {\sin ^{ - 1}}x\,dx$ and ${I_2} = \smallint {\sin ^{ - 1}}\,\,\sqrt {1\,\, - \,\,{x^2}} \;dx$, then

${I_2} = \frac{\pi }{2}\,\,{I_1}$

If $\int f(x) d x=F(x)$, then $\int x^3 f\left(x^2\right) d x$ is equal to

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

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

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Easy

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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 451-480 of 746 questions

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451 Easy

451 of 746

Application +4 | -1

\int(1-x)^{23} x d x=

View answer and explanation

Correct option: C

\frac{(1-x)^{25}}{25}-\frac{(1-x)^{24}}{24}+c

Option C is correct because it results from substituting

t = 1-x

and integrating the resulting polynomial.

452 Easy

452 of 746

Application +4 | -1

\int \frac{x+1}{\sqrt{1+x^2}} d x=

View answer and explanation

Correct option: C

\sqrt{1+x^2}+\log \left\{x+\sqrt{1+x^2}\right\}+c

Option C is correct because the integral can be split into two parts, each solved by a different method.

453 Easy

453 of 746

Application +4 | -1

\int \,\,\frac{{\cos \,\,2x}}{{\cos \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

2\,\,\sin \,\,x + \log \,\,\left( {\sec \,\,x - \,\tan x} \right) + C

Option A (

2\,\sin \,x + \log \,\left( {\sec \,x - \,\tan x} \right) + C

) is correct because after simplifying the integral using trigonometric identities and integrating, the result is

2\sin x + \log(\sec x - \tan x) + C

454 Easy

454 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\frac{{ - \,\sqrt {{x^2}\,\, + \,\,2x\,\, + \,\,2} }}{{x\,\, + \,\,1}}\,\, + \,\,C

Option C (

\frac{{ - \,\sqrt {{{{x^2}\,\, + \,\,2x\,\, + \,\,2}}} }}{{{{x\,\, + \,\,1}}}}\,\, + \,\,C

) is correct because after substitution and integration, the result matches this option.

455 Easy

455 of 746

Application +4 | -1

\int \frac{d x}{1+3 \sin ^2 x}=

View answer and explanation

Correct option: B

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

Option B is correct because the integral is solved by dividing by cos²x and using a substitution, resulting in

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

456 Easy

456 of 746

Application +4 | -1

\int \frac{1}{7+5 \cos x} d x=

View answer and explanation

Correct option: A

\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+C

Option A,

\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+C

, is correct because using the half-angle substitution and simplifying leads to this result.

457 Easy

457 of 746

Application +4 | -1

f(x)=\int\left(\frac{x^2+\sin ^2 x}{1+x^2}\right) \sec ^2 x d x

and

f(0)=0

, then

f(1)

equals:

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because evaluating the integral and applying the initial condition leads to the given result.

458 Easy

458 of 746

Application +4 | -1

For any natural number

m

\int\left(x^{7 m}+x^{2 m}+x^m\right)\left(2 x^{6 m}+7 x^m+14\right)^{1 / m} d x

, where

x>0

equals

View answer and explanation

Correct option: C

\frac{\left(2 x^{7 m}+7 x^{2 m}+14 x^m\right)^{\frac{m+1}{m}}}{14(m+1)}+C

Option C is correct because the integral can be solved by substitution, leading to the expression

\frac{\left(2 x^{7 m}+7 x^{2 m}+14 x^m\right)^{\frac{m+1}{m}}}{14(m+1)}+C

459 Easy

459 of 746

Application +4 | -1

\int g(x) d x=g(x)

, then

\int g(x)\left(f(x)-f^{\prime \prime}(x)\right) d x

is equal to

View answer and explanation

Correct option: B

g(x)\left(f(x)-f^{\prime}(x)\right)+c

Option B (

g(x)(f(x)-f^{\prime}(x))+c

) is correct because integrating by parts and using the given condition

\int g(x) dx = g(x)

simplifies the expression to this result.

460 Easy

460 of 746

Application +4 | -1

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

View answer and explanation

Correct option: D

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

Option D is correct because the integral is solved using substitution and then simplifying the resulting expression.

461 Easy

461 of 746

Application +4 | -1

\int \sqrt {1\,\, + \,\,\cos \,\,x} \,dx

is equal to

View answer and explanation

Correct option: B

2\,\,\sqrt 2 \,\,\sin \,\,\frac{x}{2}\,\, + \,\,c

Option B,

2\,\,\sqrt 2 \,\,\sin \,\,\frac{x}{2}\,\, + \,\,c

, is correct because using trigonometric identities, the integral can be simplified and solved, resulting in this expression.

462 Easy

462 of 746

Application +4 | -1

\int(x+1)^2 e^x d x

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because after expanding and integrating by parts, the integral simplifies to this expression.

463 Easy

463 of 746

Application +4 | -1

The value of the integral

\int e^{\sin ^2 x}\left(\cos x+\cos ^3 x\right) \sin x d x

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because after substituting

t=\sin^2 x

and integrating, we obtain the given result.

464 Easy

464 of 746

Application +4 | -1

\int \frac{\cos 2 x+x+1}{x^2+\sin 2 x+2 x} d x=

View answer and explanation

Correct option: C

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

Option C is correct because the integral is solved by recognizing the numerator as half the derivative of the denominator, leading to a logarithmic solution.

465 Easy

465 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 is correct because after performing the integration and substituting back, the value of k in s-z = k/x is found to be 2.

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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

466 Easy

466 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

2 \sqrt{1+\sqrt{1+x^2}}+C

Option D is correct because the integral simplifies to

2 \sqrt{1+\sqrt{1+x^2}}+C

after a suitable substitution.

467 Easy

467 of 746

Application +4 | -1

\int \frac{x+x^{2 / 3}+x^{1 / 6}}{x\left(1+x^{1 / 3}\right)} d x

equals to (Where

{C}

is constant of integration)

View answer and explanation

Correct option: D

\frac{3}{2} x^{2 / 3}+6 \tan ^{-1} x^{1 / 6}+C

Option D (3/2 x^(2/3) + 6 tan^-1 x^(1/6) + C) is correct because by substituting x^(1/6) = t, the integral simplifies and can be evaluated to this expression.

468 Easy

468 of 746

Application +4 | -1

I = \int {\frac{1}{{2p}}\sqrt {\frac{{p - 1}}{{p + 1}}} dp = f\left( p \right) + c,}

then

f\left( p \right)

is equal to

View answer and explanation

Correct option: C

\ln \sqrt {p + \sqrt {{p^2} - 1} } - \frac{1}{2}{\sec ^{ - 1}}p

Option D (None of these) is incorrect. Option C (

\ln \sqrt {p + \sqrt {{p^2} - 1} } - \frac{1}{2}{\sec ^{ - 1}}p

) is correct because after simplifying the integral, it can be split into two parts, each integrating to the terms in option C.

469 Easy

469 of 746

Application +4 | -1

\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x-\cos x}{b}\right)+C

, then

(a, b)

is equal to

View answer and explanation

Correct option: A

(1,3)

Option A,

(1,3)

, is correct because manipulating the integrand and completing the square leads to this result.

470 Easy

470 of 746

Understanding +4 | -1

Let

f

and

g

be two polynomials then

\int\left(f(x) g^{\prime \prime}(x)-f^{\prime \prime}(x) g(x)\right) \mathrm{d} x

is equal to (ignoring the constant of integration)

View answer and explanation

Correct option: C

f(x) g^{\prime}(x)-f^{\prime}(x) g(x)

Option C (

f(x) g^{\prime}(x)-f^{\prime}(x) g(x)

) is correct because it is the result of applying integration by parts twice to the given integral.

471 Easy

471 of 746

Application +4 | -1

\int \,\,\left[ {{e^{a\,\,\log \,\,x}}\,\, + \,\,{e^{x\,\,\log \,\,a}}} \right]\,\,dx\,\, = ... + \,\,c\,\,;\,\,a,\,\,x\,\, > 1

View answer and explanation

Correct option: A

\frac{{{x^{a + 1}}}}{{a + 1}}\,\, + \,\,\frac{{{a^x}}}{{\log \,\,a}}

Option A is correct because the integral of the given expression simplifies to the sum of a power function and an exponential function, which matches option A.

472 Easy

472 of 746

Analyzing +4 | -1

{I_1} = \int {\sin ^{ - 1}}x\,dx

and

{I_2} = \int {\sin ^{ - 1}}\,\,\sqrt {1\,\, - \,\,{x^2}} \;dx

, then

View answer and explanation

Correct option: C

{I_1} + \,\,{I_2} = \frac{\pi }{2}\,\,x

Option C (

{I_1} + \, {I_2} = \frac{\pi }{2}\,x

) is correct because the sum of the integrals simplifies to an integral involving the constant

\frac{\pi}{2}

473 Easy

473 of 746

Application +4 | -1

\int f(x) d x=F(x)

, then

\int x^3 f\left(x^2\right) d x

is equal to

View answer and explanation

Correct option: B

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

Option B (

\frac{1}{2}[x^2 F(x^2)-\int F(x^2) d(x^2)]

) is correct because it is the result of applying integration by parts to the integral

\int x^3 f(x^2) dx

with a suitable substitution.

474 Easy

474 of 746

Analyzing +4 | -1

\int \frac{x^2-1}{x \sqrt{x^4+3 x^2+1}} d x

is equal to

View answer and explanation

Correct option: A

\log _e\left|x+\frac{1}{x}+\sqrt{x^2+\frac{1}{x^2}+3}\right|+C

Option A (

\log _e\left|x+\frac{1}{x}+\sqrt{x^2+\frac{1}{x^2}+3}\right|+C

) is correct because after dividing by

x^2

and completing the square, the integral can be solved using a standard logarithmic form.

475 Easy

475 of 746

Application +4 | -1

\int e^x \sin x d x=

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of applying integration by parts twice to the integral of

e^x \sin x

476 Easy

476 of 746

Analyzing +4 | -1

The integral

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

is equal to
where,

C

is an arbitrary constant.

View answer and explanation

Correct option: B

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

Option B is correct because after substitution and simplification, the integral evaluates to the given expression.

477 Easy

477 of 746

Application +4 | -1

The value of

\int \frac{\sqrt{1+x}}{x} d x

, is

View answer and explanation

Correct option: A

2 \sqrt{1+x}+\log \left|\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}\right|+C

Option A is correct because the integral is solved using a substitution method, resulting in the given expression with a logarithmic term.

478 Easy

478 of 746

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{1}{n}\,\,\log \,\,\left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right)\,\, + \,\,C

Option A (

\frac{1}{n}\,\,\log \,\,\left( {\frac{{{x^n}}}{{{x^n} + 1}}} \right)\, + \,C

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

479 Easy

479 of 746

Understanding +4 | -1

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

View answer and explanation

Correct option: C

\frac{e^x}{x}+c

Option C (

\frac{{e^x}}{{x}}+c

) is correct because the integral is of the form

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

, where

f(x) = \frac{{1}}{{x}}

and

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

480 Easy

480 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

\tan (1+\log x)+c

Option A is correct because the integral simplifies to

\tan(1 + \log x) + c

after applying the substitution method.

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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