$\smallint \,\,\frac{{10\,\,{x^9} + \,\,{{10}^x}\,\,{{\log }_e}10}}{{{{10}^x} + {x^{10}}}}\,\,dx$ is equal to

${10^x}\,\, - \,\,{x^{10}} + \,\,c$

${10^x}\,\, + \,\,{x^{10}} + \,\,c$

${\left( {{{10}^x}\,\, - \,\,{x^{10}}} \right)^{ - 1}} + \,\,c$

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

$\frac{{ - \,{e^x}}}{{x\,\, + \,\,1}}\,\, + \,\,c$

$\frac{{{e^x}}}{{x\,\, + \,\,1}}\,\, + \,\,c$

$\frac{{x{e^x}}}{{x\,\, + \,\,1}}\,\, + \,\,c$

The value of $\smallint \,\,\frac{{dx}}{{\sqrt x \,\, + \,\,\sqrt[3]{x}}}$ is

$3\,\sqrt x \,\, + \,\,3\,\,\left( {\sqrt[3]{x}} \right)\,\, - \,\,6\,\;\sqrt[6]{x}\,\, + \,\,6\,\,\log \,\,\left( {\sqrt[6]{{x + 1}}} \right)\,\, + \,\,c$

$2\,\sqrt x \,\, + \,\,6\,\,\left( {\sqrt[6]{x}} \right)\,\, - \,\,6\,\,\log \,\,\left( {\sqrt[6]{{x\,\, + \,\,1}}} \right)\,\, + \,\,c$

$\smallint \,\,\frac{{{{\sin }^{ - 1}}x}}{{\sqrt {1 - {x^2}} }}\;\,\,dx$ is equal to

$\log \,\,({\sin ^{ - 1}}x) + c$

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

$\smallint \,\,\frac{{\sin \,\,x\,\,\cos \,\,x}}{{\sqrt {1\,\, - \,\,{{\sin }^4}\,\,x} }}\,\,dx$ is equal to

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

${\tan ^{ - 1}}\,\,\left( {{{\sin }^2}\,\,x} \right) + c$

${\tan ^{ - 1}}\,\,\left( {2\,\,{{\sin }^2}\,\,x} \right) + c$

The antiderivative of $\frac{1}{\sin ^2 x+\tan ^2 x}$ is

$(-1 / 2)\left[\tan x+(1 / \sqrt{2}) \tan ^{-1}(1 / \sqrt{2}) \tan x\right]+C$

The value of $\smallint \,\,\frac{{{{\cos }^3}\,\,x\,\, + \,\,{{\cos }^5}\,\,x}}{{{{\sin }^2}\,\,x\,\, + \,\,{{\sin }^4}\,\,x}}\,\,dx$, is

$\sin \,\,x\,\, - \,\,6\,\,{\tan ^{ - 1}}\,\,\left( {\sin \,\,x} \right) + C$

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

$\sin \,\,x\,\, - \,\,2\,\,{\left( {\sin \,\,x} \right)^{ - 1}}\,\, + \,\,5\,\,{\tan ^{ - 1}}\,\,\left( {\sin \,\,x} \right)\,\, + \,\,C$

If $\int\left\{\log (\log x)+\frac{1}{(\log x)^2}\right\} d x=x\{f(x)-g(x)\}+C,$ then

$f(x)=\log x ; g(x)=\frac{1}{\log x}$

$f(x)=\frac{1}{\log x} ;(x)=\log (\log x)$

$f(x)=\frac{1}{x \log x} ; g(x)=\frac{1}{\log x}$

$\smallint {e^x}\,\,\left( {\frac{{1 - \,\,\sin \,\,x}}{{1 - \,\,\cos \,\,x}}} \right)\;dx$ is equal to

$ - \,{e^x}\,{\rm{cot}}\,\,\left( {\frac{x}{2}} \right) + c$

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

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

$\smallint \,\,\frac{{m{x^{m + 2n - 1}}\,\, - \,\,n{x^{n - 1}}}}{{{x^{2m + 2n}}\,\, + \,\,2{x^{m + n}}\,\, + \,\,1}}\;\,\,dx$ is equal to

$\frac{{{x^m}}}{{{x^{m + n}}\,\, + \,\,1}}\,\, + \,\,c$

$\frac{{{x^{m + n}}\,\, - \,\,1}}{{{x^{m + n}}\,\, + \,\,1}}\,\, + \,\,c$

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

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

$c\,\, - \,\,{e^x}\,\,\cot \left( {\frac{x}{2}} \right)$

If $I = \smallint \,\,\frac{{{x^5}}}{{\sqrt {1\,\, + \,\,{x^3}} }}\,\,dx$, then $I$ is equal to

$\frac{2}{9}\,\,{\left( {1 + {x^3}} \right)^{\frac{5}{2}}}\,\, + \,\,\frac{2}{3}\,\,{\left( {1 + {x^3}} \right)^{\frac{3}{2}}}\,\, + \,\,c$

$\log \,\,\left| {\,\,\sqrt x \,\, + \,\,\sqrt {1 + {x^3}} \,} \right|\,\, + \,\,c$

$\log \,\,\left| {\,\sqrt x \,\, - \,\,\sqrt {1 - {x^3}} \,} \right| + \,\,c$

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Mixed difficulty

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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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301 Easy

301 of 995

Application +4 | -1

\int \,\,\frac{{\sqrt x }}{{x + 1}}\,\,dx = A\,\sqrt x + B\,\,{\tan ^{ - 1}}\,\,\sqrt x + c

, then

View answer and explanation

Correct option: D

A = 2,\,\,B = - 2

Option D is correct because after performing the substitution and integration, the values of A and B are found to be 2 and -2 respectively.

302 Easy

302 of 995

Understanding +4 | -1

\int \,\,\frac{{10\,\,{x^9} + \,\,{{10}^x}\,\,{{\log }_e}10}}{{{{10}^x} + {x^{10}}}}\,\,dx

is equal to

View answer and explanation

Correct option: D

\log \,\,\left( {{{10}^x}\,\, + \,\,{x^{10}}} \right)\,\, + \,\,c

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

303 Easy

303 of 995

Application +4 | -1

\int \frac{e^x(1+\sin x)}{1+\cos x} \mathrm{~d} x

is equal to

View answer and explanation

Correct option: B

e^x \tan x / 2+C

Option B (

e^x \tan x / 2+C

) is correct because the integral simplifies to the form

\int e^x (f(x) + f'(x)) dx = e^x f(x) + C

, where

f(x) = \tan(x/2)

304 Easy

304 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

{e^x}\,\,\frac{{x\,\, - \,1}}{{x\,\, + \,\,1}}\,\, + \,\,c

Option C (

{e^x}\frac{{x\\ - \\,1}}{{x\\ + \\,1}}\\ + \\,c

) is correct because the integral can be simplified and solved using integration techniques.

305 Easy

305 of 995

Application +4 | -1

The value of

\int \,\,\frac{{dx}}{{\sqrt x \,\, + \,\,\sqrt[3]{x}}}

View answer and explanation

Correct option: C

2\,\sqrt x \,\, - \,\,3\,\,\left( {\sqrt[3]{x}} \right)\,\, + \,\,6\,\;\left( {\sqrt[6]{x}} \right)\;\, - \,6\,\,\log \,\,\left( {\sqrt[6]{{x + 1}}} \right)\,\, + \,\,c

Option C (

2\,\sqrt x \, - \,3\,\left( {\sqrt[3]{x}} \right)\, + \,6\;\left( {\sqrt[6]{x}} \right)\, - \,6\,\log \,\left( {\sqrt[6]{{x + 1}}} \right)\, + \,c

) is correct because it is the result of a u-substitution and polynomial long division.

306 Medium

306 of 995

Understanding +4 | -1

f'\,\,\left( x \right) = x\,\, + \,\,\frac{1}{x}

, then the value of

f\left( x \right)

View answer and explanation

Correct option: B

\frac{{{x^2}}}{2}\,\, + \,\,\log \,\,x\,\, + \,\,c

Option B,

\frac{{{x^2}}}{2}\,+\,\log \,x\,+\,c

, is correct because it is the result of integrating

x + \frac{1}{x}

307 Easy

307 of 995

Application +4 | -1

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

equals

View answer and explanation

Correct option: D

None of these

None of the options are correct. The correct answer is

\frac{2}{\sqrt{7}} \tan^{-1} \frac{(4x+1)}{\sqrt{7}}+C

308 Easy

308 of 995

Application +4 | -1

\int \frac{1}{\cos x+\sqrt{3} \sin x} d x

equals

View answer and explanation

Correct option: C

\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+C

Option C,

\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+C

, is correct because rewriting the integrand using trigonometric identities and integrating leads to this result.

309 Easy

309 of 995

Application +4 | -1

\int x^2 \sin 2 x d x=

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the result of integrating

x^2 \sin 2x

twice using integration by parts.

310 Easy

310 of 995

Application +4 | -1

The value of

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

, is

View answer and explanation

Correct option: C

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

Option C,

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

, is correct because using the substitution

x-1=t^2

and completing the square leads to this result.

311 Medium

311 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

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

Option B (

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

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

312 Medium

312 of 995

Analyzing +4 | -1

\int \sqrt{(x-3)}\left[\sin ^{-1}(\ln x)+\cos ^{-1}(\ln x)\right] d x

equals

View answer and explanation

Correct option: C

does not exist

Option C is correct because the domains of the two functions inside the integral do not overlap, so the integral does not exist.

313 Easy

313 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it results from a straightforward u-substitution with

u = \sin^{-1}x

314 Easy

314 of 995

Analyzing +4 | -1

\int \,\,\frac{{2{x^2}\,\, + \,\,3}}{{\left( {{x^2} - 1} \right)\,\,\left( {{x^2} + 4} \right)}}\,\,dx = a\,\,\log \,\,\left( {\frac{{x + 1}}{{x - 1}}} \right)\,\, + \,\,b\,\,{\tan ^{ - 1}}\,\,\frac{x}{2}

, then

\left( {a,\,\;b} \right)

View answer and explanation

Correct option: A

\left( { - 1/2,\;\,\,1/2} \right)

Option A ((-1/2, 1/2)) is correct because by performing partial fraction decomposition and integrating, we find the coefficients a and b to be -1/2 and 1/2, respectively.

315 Easy

315 of 995

Application +4 | -1

\int \,\,\frac{{\sin \,\,x\,\,\cos \,\,x}}{{\sqrt {1\,\, - \,\,{{\sin }^4}\,\,x} }}\,\,dx

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because after substitution, the integral simplifies to a standard arcsin form.

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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

316 Easy

316 of 995

Application +4 | -1

The value of

\int \frac{\sin x+\cos x}{3+\sin 2 x} d x

, is

View answer and explanation

Correct option: A

\frac{1}{4} \log \left(\frac{2+\sin x-\cos x}{2-\sin x+\cos x}\right)+C

Option A is correct because the integral of the given function can be solved by manipulating the denominator and using substitution, leading to a logarithmic form.

317 Easy

317 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because the integral simplifies to (tan^(-1)(x))⁴ / 4 + C after recognizing the derivative of tan^(-1)(x).

318 Easy

318 of 995

Application +4 | -1

The antiderivative of

\frac{1}{\sin ^2 x+\tan ^2 x}

View answer and explanation

Correct option: B

(-1 / 2)\left[\cot x+(1 / \sqrt{2}) \tan ^{-1}((1 / \sqrt{2}) \tan x)\right]+C

Option B is the correct antiderivative, obtained by simplifying the integrand, substituting \

t = \tan x

and integrating the resulting expression.

319 Easy

319 of 995

Application +4 | -1

Let

f

be a function such that

f\left( 1 \right) = 4

and

f'\left( x \right) \ge 2

for

1 \le x < 4

. How small can

f\left( 4 \right)

possibly be?

View answer and explanation

Correct option: D

The correct answer is D (10) because it represents the smallest possible value of f(4) based on the given conditions and the lower bound of the derivative.

320 Easy

320 of 995

Analyzing +4 | -1

Let

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

and

f(0)=0

. Then

f(1)

is equal to

View answer and explanation

Correct option: B

\log _e(1+\sqrt{2})-\frac{\pi}{4}

Option B (

\log _e(1+\sqrt{2})-\frac{\pi}{4}

) is correct because it is the result of evaluating the definite integral with the given initial condition.

321 Easy

321 of 995

Analyzing +4 | -1

The value of

\int \,\,\frac{{{{\cos }^3}\,\,x\,\, + \,\,{{\cos }^5}\,\,x}}{{{{\sin }^2}\,\,x\,\, + \,\,{{\sin }^4}\,\,x}}\,\,dx

, is

View answer and explanation

Correct option: C

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

Option C is correct because the integral simplifies to

\sin x - 2(\sin x)^{-1} - 6\tan^{-1}(\sin x) + C

after substitution and partial fraction decomposition.

322 Easy

322 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\log |\sin \theta+\cos \theta|+c

The correct answer is B (3) because by using the given conditions and trigonometric identities, we can derive that cot(alpha)cot(gamma) = 3.

323 Easy

323 of 995

Application +4 | -1

\int\left\{\log (\log x)+\frac{1}{(\log x)^2}\right\} d x=x\{f(x)-g(x)\}+C,

then

View answer and explanation

Correct option: A

f(x)=\log (\log x) ; g(x)=\frac{1}{\log x}

Option A (

f(x)=\log (\log x) ; g(x)=\frac{1}{\log x}

) is correct because after integration by parts and substitution, the given integral matches the form

x\{f(x)-g(x)\} + C

324 Medium

324 of 995

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 by using the trigonometric identity

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

, the integral simplifies to

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

325 Medium

325 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because after simplifying the integrand and applying integration by parts, the result matches this option.

326 Medium

326 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (-

(1+\frac{1}{x^4})^{1 / 4}+c

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

327 Easy

327 of 995

Analyzing +4 | -1

\int \,\,\frac{{\cos \,\,4x\,\, + \,\,1}}{{\cot \,\,x\,\, - \,\,\tan \,\,x}}\,\,dx = k\,\,\cos \,\,4x + \,\,c

, then

View answer and explanation

Correct option: B

k = - 1/8

Option B (

k = - 1/8

) is correct because after simplifying and integrating the given expression, the coefficient of

\cos 4x

is -1/8.

328 Easy

328 of 995

Analyzing +4 | -1

\int \,\,\frac{{m{x^{m + 2n - 1}}\,\, - \,\,n{x^{n - 1}}}}{{{x^{2m + 2n}}\,\, + \,\,2{x^{m + n}}\,\, + \,\,1}}\;\,\,dx

is equal to

View answer and explanation

Correct option: D

- \,\,\frac{{{x^n}}}{{{x^{m + n}}\,\, + \,\,1}}\,\, + \,\,c

Option D (

- \,\frac{{{x^n}}}{{{{x^{m + n}}\\ + \\,1}}}\\ + \\,c

) is correct because it is the result of the integration after appropriate substitution and simplification.

329 Easy

329 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

{e^x}\,\,\tan \,\,\left( {\frac{x}{2}} \right) + c

Option A is correct because the integral can be simplified and solved using the properties of exponential functions and trigonometric identities.

330 Medium

330 of 995

Application +4 | -1

I = \int \,\,\frac{{{x^5}}}{{\sqrt {1\,\, + \,\,{x^3}} }}\,\,dx

, then

I

is equal to

View answer and explanation

Correct option: D

\frac{2}{9}\,\,{\left( {1 + {x^3}} \right)^{\frac{3}{2}}}\,\, - \,\,\frac{2}{3}\,\,{\left( {1 + {x^3}} \right)^{\frac{1}{2}}}\,\, + \,\,c

Option D (

\frac{2}{9}\,{\left( {1 + {x^3}} \right)^{\frac{3}{2}}}\, - \,\frac{2}{3}\,{\left( {1 + {x^3}} \right)^{\frac{1}{2}}}\, + \,c

) is correct because it results from a correct u-substitution and integration.

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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