If $\int \frac{d x}{x \sqrt{5 x^2-3}}=K \tan ^{-1} f(x)+C$ then

$f(x)=\sqrt{\frac{5}{3} x^2-1}, K=\frac{1}{\sqrt{5}}$

$f(x)=\frac{1}{2} \sqrt{5 x^2-3}, K=\frac{1}{\sqrt{5}}$

If $f\left(\frac{3 x-4}{3 x+4}\right)=x+2$, then $\int f(x) d x$ is:

$e^{x+2} \log \left|\frac{3 x-4}{3 x+4}\right|+c$

$\frac{8}{3} \log |1-x|+\frac{x}{3}+c$

$e^{[(3 x-4)(3 x+4)]}-\frac{x^2}{2}-2 x+c$

$\smallint {5^{{5^{{5^x}}}}} \cdot \,\,{5^{{5^x}}} \cdot \,\,{5^x}\;dx$ is equal to

$\frac{{{5^{{5^x}}}}}{{{{\left( {\log \,\,5} \right)}^3}}}\,\, + \,\,C$

The integral $\smallint \frac{{2x\,\, - \,\,3}}{{{{\left( {{x^2} + \,\,x\,\, + 1} \right)}^2}}}\,\,dx$ is equal to

$ - \,\,\frac{1}{{{x^2} + x + 1}}\,\, - \,\,\frac{4}{3}\,\,{\tan ^{ - 1}}\,\,\left( {4x + 3} \right) + C$

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

$\frac{1}{{4\left( {{x^2} + x + 1} \right)}}\,\, + \,\,\frac{2}{3}\,\,{\tan ^{ - 1}}\,\,\left( {2x + 1} \right) + C$

If $\smallint \,\,\frac{1}{{x + {x^5}}}\,\,dx = f\left( x \right) + c$, then the value of $\smallint \frac{{{x^4}}}{{x + {x^5}}}\,\;dx$ equals

$f\left( x \right)\,\, + \,\,\log \,\,x\,\, + \,\,c$

$f\left( x \right)\,\, - \,\,\log \,\,x\,\, + \,\,c$

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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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511 Easy

511 of 746

Application +4 | -1

\int \frac{d x}{x \sqrt{5 x^2-3}}=K \tan ^{-1} f(x)+C

then

View answer and explanation

Correct option: B

f(x)=\sqrt{\frac{5}{3} x^2-1}, K=\frac{1}{\sqrt{3}}

Option B (

f(x)=\sqrt{{\frac{{5}}{{3}} x^2-1}}, K=\frac{{1}}{{\sqrt{{3}}}}

) is correct because the integral can be solved by substitution, leading to an arctangent integral.

512 Easy

512 of 746

Application +4 | -1

\int\left(1+x-x^{-1}\right) e^{x+x^{-1}} d x

, is equal to

View answer and explanation

Correct option: A

x e^{x+x^{-1}}+C

Option A is correct because the integral simplifies to

x e^{x+x^{-1}}+C

after applying integration by parts.

513 Easy

513 of 746

Analyzing +4 | -1

The value of

\int \frac{d x}{\cos ^6 x+\sin ^6 x}

, is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because after simplifying the integral using trigonometric identities and substitution, we arrive at this result.

514 Easy

514 of 746

Application +4 | -1

The value of

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

, is

View answer and explanation

Correct option: B

-\left(1+\frac{1}{x^4}\right)^{1 / 4}

Option B is correct because the integral simplifies to -(1 + 1/x⁴)^(1/4) + C after substitution and simplification.

515 Easy

515 of 746

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B is correct because substituting

t = x^2

transforms the integral to

\frac{1}{2} \int \frac{1}{1 + t^2} dt

, which equals

\frac{1}{2} \tan^{-1} t + c = \frac{1}{2} \tan^{-1} x^2 + c

516 Easy

516 of 746

Application +4 | -1

Let

x^2 \neq n \pi-1, n \in N

. Then, the value of

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

is equal to

View answer and explanation

Correct option: B

\log \left|\sec \left(\frac{x^2+1}{2}\right)\right|+C

Option B is correct because after simplifying the expression inside the square root and using a trigonometric identity, the integral can be evaluated as

\log \left|\sec \left(\frac{x^2+1}{2}\right)\right|+C

517 Easy

517 of 746

Application +4 | -1

\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x

is equal to

View answer and explanation

Correct option: D

2 \sin \sqrt{x}+c

Option D is correct because substituting

t = \sqrt{x}

simplifies the integral to

2\int \cos t dt

, which integrates to

2\sin \sqrt{x} + c

518 Easy

518 of 746

Application +4 | -1

\int \frac{\sqrt{5+x^{10}}}{x^{16}} d x=a\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+C

, then

a=

View answer and explanation

Correct option: C

-\frac{1}{75}

Option C is correct because after substitution and integration, the value of 'a' is found to be -1/75.

519 Easy

519 of 746

Application +4 | -1

\int \frac{d x}{\cos ^6 x+\sin ^6 x}

is equal to

View answer and explanation

Correct option: C

\operatorname{tar}^{21}(\tan x-\cot x)+c

Option C (

\operatorname{tar}^{21}(\tan x-\cot x)+c

) is correct because after simplification and substitution, the integral results in the arctangent function.

520 Easy

520 of 746

Application +4 | -1

\int \frac{\mathrm{d} x}{a^2 \cos ^2 x+b^2 \sin ^2 x}(a, b>0)

is equal to

View answer and explanation

Correct option: C

\frac{1}{a b} \tan ^{-1}\left(\left(\frac{b}{a}\right) \tan x\right)+C

Option C (

\frac{{1}}{{a b}} \tan ^{-1}\left(\left(\frac{{b}}{{a}}\right) \tan x\right)+C

) is correct because dividing by

\cos^2 x

and substituting

t = \tan x

leads to an arctangent integral.

521 Easy

521 of 746

Application +4 | -1

f\left(\frac{3 x-4}{3 x+4}\right)=x+2

, then

\int f(x) d x

is:

View answer and explanation

Correct option: B

-\frac{8}{3} \log |1-x|+\frac{2}{3} x+c

Option B is correct because the integral can be solved by substitution and partial fractions.

522 Easy

522 of 746

Application +4 | -1

\int \frac{f(x)}{x^2-x+1} \mathrm{~d} x=\frac{3}{2} \log \left(x^2-x+1\right)+

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

then

f(x)

is equal to

View answer and explanation

Correct option: C

3 x-1

The correct answer is

3x-1

because differentiating the given integral and simplifying leads to

f(x) = 3x - 1

523 Easy

523 of 746

Application +4 | -1

\int e^x(1+x) \sec ^2\left(x e^x\right) d x=f(x)+

Constant, then

f(x)

is equal to

View answer and explanation

Correct option: D

\tan \left(x e^x\right)

Option D (

\tan \left(x e^x\right)

) is correct because the integral simplifies to

\tan(xe^x) + C

, so

f(x) = \tan(xe^x)

524 Easy

524 of 746

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 is correct because differentiating both sides of the equation and solving for f(4/25) yields 2/5.

525 Easy

525 of 746

Application +4 | -1

{I_n} = \int \,\,{x^n}\,.\,\,{e^{cx}}\;\,dx\;\;\,\mathrm{for}\,\;\;n\,\, \ge \,\,1,\,\;\mathrm{then}\,\,c\,.\,{I_n}\,\, + \,\,n\,.\,{I_{n - 1}}

is equal to

View answer and explanation

Correct option: A

{x^n}\,{e^{cx}}

Option A (

{x^n}\,{e^{cx}}

) is correct because integrating by parts and rearranging the terms leads to 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

526 Easy

526 of 746

Analyzing +4 | -1

\int \frac{x^4+1}{x^6+1} d x

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because by factoring the denominator and simplifying the integrand, the integral can be expressed as the sum of two standard integrals.

527 Easy

527 of 746

Application +4 | -1

\int \sin ^3 x \cdot \cos x d x=

View answer and explanation

Correct option: B

\frac{\sin ^4 x}{4}+c

Option B (

\frac{\sin ^4 x}{4}+c

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

528 Easy

528 of 746

Application +4 | -1

\int \log x(\log x+2) d x=

View answer and explanation

Correct option: A

x(\log x)^2+c

Option A (

x(\log x)^2+c

) is correct because the integral evaluates to this expression after substitution and simplification.

529 Easy

529 of 746

Application +4 | -1

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

, then

View answer and explanation

Correct option: A

A=1 / \sqrt{5}

Option A (

A=1 / \sqrt{5}

) is correct because after substitution and integration, the coefficient of the inverse tangent term is

1/\sqrt{5}

530 Easy

530 of 746

Application +4 | -1

\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\sin \mathrm{x}}{1-\cos \mathrm{x}}\right) \mathrm{dx}

is equal to

View answer and explanation

Correct option: B

-\mathrm{e}^{\mathrm{x}} \cot \left(\frac{\mathrm{x}}{2}\right)+\mathrm{C}

Option B (

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

) is correct because it is the result of integrating the given expression after simplification using trigonometric identities.

531 Easy

531 of 746

Analyzing +4 | -1

\int {5^{{5^{{5^x}}}}} \cdot \,\,{5^{{5^x}}} \cdot \,\,{5^x}\;dx

is equal to

View answer and explanation

Correct option: C

\frac{{{5^{{5^{{5^x}}}}}}}{{{{\left( {\log \,\,5} \right)}^3}}}\,\, + \,\,C

Option A (

\mathrm{1}

) is correct because the limit is evaluated by simplifying the integral and applying the limit definition.

532 Easy

532 of 746

Analyzing +4 | -1

\int {\frac{{\cos \theta }}{{5 + 7\sin \theta - 2{{\cos }^2}\theta }}} \,d\theta \,\, = A\,\,{\log _e}\left| {B(\theta )} \right| + C,

where

C

is constant of integration, then

\frac{{B(\theta )}}{A}

can be

View answer and explanation

Correct option: D

\frac{{5(2\sin \theta + 1)}}{{\sin \theta + 3}}

Option D is correct because after integrating the given expression and simplifying, the result matches the form

A \log_e |B(\theta)| + C

, leading to the correct value of

\frac{{B(\theta)}}{A}

533 Easy

533 of 746

Application +4 | -1

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

View answer and explanation

Correct option: A

-\frac{1}{(x+1)}+c

Option A (

-\frac{{1}}{{(x+1)}}+c

) is correct because the integral simplifies to this form after simplifying the fraction.

534 Easy

534 of 746

Analyzing +4 | -1

The integral

\int \frac{{2x\,\, - \,\,3}}{{{{\left( {{x^2} + \,\,x\,\, + 1} \right)}^2}}}\,\,dx

is equal to

View answer and explanation

Correct option: A

- \,\,\frac{{8x + 7}}{{{x^2} + x + 1}}\,\, - \,\,\frac{{16}}{{3\,\sqrt 3 }}\,\,{\tan ^{ - 1}}\,\,\left( {\frac{{2x + 1}}{3}} \right) + C

Option A is the correct answer because the integral can be solved by breaking it into parts and using trigonometric substitution to evaluate the remaining integral.

535 Easy

535 of 746

Application +4 | -1

\int \left( {\sqrt[3]{x}} \right)\,\,\left( {\sqrt[5]{{1\,\, + \,\,\sqrt[3]{{{x^4}}}}}} \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{5}{8}\,\,{\left( {1\,\, + \,\,{x^{\frac{4}{3}}}} \right)^{\frac{6}{5}}}\,\, + \,\,c

Option C (

\frac{5}{8}\left( {1\\,\\ + \\,{x^{\frac{{4}}{{3}}}}} \right)^{\frac{{6}}{{5}}}}\\ + \\,c

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

536 Easy

536 of 746

Application +4 | -1

\int \frac{d x}{1+e^x}=

View answer and explanation

Correct option: B

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

Option B is correct because the integral can be rewritten and solved using a substitution, resulting in

-\log(1+e^{-x})

537 Easy

537 of 746

Application +4 | -1

\int \,\,\frac{1}{{x + {x^5}}}\,\,dx = f\left( x \right) + c

, then the value of

\int \frac{{{x^4}}}{{x + {x^5}}}\,\;dx

equals

View answer and explanation

Correct option: A

\log \,\,x\,\, - \,\,f\,\,\left( x \right)\,\, + \,\,c

Option A (

\log \,\,x\,\, - \,\,f\,\,\left( x \right)\,\, + \,\,c

) is correct because it is obtained by manipulating the integrand and using the given integral.

538 Easy

538 of 746

Application +4 | -1

\int \frac{1+\cos 8 x}{\tan 2 x-\cot 2 x} d x=a \cos 8 x+C

, then

a=

View answer and explanation

Correct option: C

\frac{1}{16}

Option C (

\frac{1}{16}

) is correct because after simplifying the integral, the coefficient of

\cos 8x

\frac{1}{16}

539 Easy

539 of 746

Application +4 | -1

\int 3^{3^{3^x}} 3^{3^x} 3^x \mathrm{~d} x

is equal to

View answer and explanation

Correct option: C

\frac{3^{3^{3^5}}}{(\log 3)^3}+C

Option C is correct because after applying the substitution

t = 3^{3^{3^x}}

, the integral simplifies to

\frac{3^{3^{3^x}}}{(\log 3)^3}+C

540 Easy

540 of 746

Application +4 | -1

The integral

\int \frac{d x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}

equals

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it represents the result of the given integral after applying substitution and simplification.

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