The value of the integral $\smallint \,\,{e^x}\,\,{\left( {\frac{{1 - x}}{{1 + x}}} \right)^2}\,dx$ is

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

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

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

$\mathop \smallint \nolimits_0^3 \,\,\frac{{3x\,\, + \,\,1}}{{{x^2}\,\, + \,\,9\;}}\;\,\,dx$ is equal to

$\log \,\,\left( {2\,\sqrt 2 } \right)\,\, + \,\,\frac{\pi }{2}$

$\log \,\,\left( {2\,\,\sqrt 2 } \right)\,\, + \,\,\frac{\pi }{6}$

$\log \,\,\left( {2\,\,\sqrt 2 } \right)\,\, + \,\,\frac{\pi }{3}$

$\smallint \,\,\frac{{{x^2}\,\, - \,\,1}}{{{x^4}\,\, + \,\,{x^2}\,\, + \,\,1}}\,\,dx$ is equal to

$\log \,\,\left( {{x^4}\,\, + \,\,{x^2}\,\, + \,\,1} \right)\,\, + \,\,c$

If $I = \smallint \,\,\frac{1}{{{x^4}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} }}\,\,dx$, then I equals

$\frac{1}{{{a^4}}}\,\,\left\{ {\frac{1}{x}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} \,\, - \,\,\frac{1}{{3{x^3}}}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} } \right\} + C$

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

$\smallint \,\,\frac{{\sin \,\,x}}{{\sin \,(x\,\, - \,\,\alpha )}}\,\,dx$ is

$x\,\,\sin \,\,\alpha \,\, + \,\,\cos \,\,\alpha \,\,\log \,\,\sin \,\,(x + \alpha )\,\, + \,\,c$

$x\,\,\sin \,\,\alpha \,\, + \,\,\cos \,\,\alpha \,\,\log \,\,\sin \,\,(x - \alpha )\,\, + \,\,c$

$x\,\,\cos \,\,\alpha \,\, + \,\,\sin \,\,\alpha \,\,\log \,\,\cos \,\,(x + \alpha )\,\, + \,\,c$

$\mathop \smallint \nolimits_0^\lambda \,\,\frac{{y\,\,dy}}{{\sqrt {y + \lambda } }}$ is equal to

$\frac{1}{3}\,\,\left( {2 - \sqrt 2 } \right)\,\,\lambda \,\,\sqrt \lambda $

$\frac{1}{3}\,\,\left( {2 + \sqrt 2 } \right)\,\,\lambda \,\,\sqrt \lambda $

If $\smallint \,\,{\cos ^4}\,x\,\,dx = Ax\,\, + \,\,B\,\,\sin \,\,2\,x\,\, + \,\,C\,\,\sin \,\,4x\,\, + \,\,D$ , then {A,B,C} equals

$\left\{ {\frac{3}{8},\,\,\frac{1}{{32}}\,\,,\,\,\frac{1}{4}} \right\}$

$\left\{ {\frac{1}{{32}},\,\,\frac{1}{4},\,\,\frac{3}{8}} \right\}$

$\left\{ {\frac{1}{4},\,\,\frac{3}{8},\,\,\frac{1}{{32}}} \right\}$

$\smallint \,{\left( {\frac{{x + 2}}{{x + 4}}} \right)^2}{e^x}\,dx$ is equal to

${e^x}\,\,\left( {\frac{{x\,\, + \,\,2}}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c$

${e^x}\,\,\left( {\frac{{x\,\, - \,\,2}}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c$

$\left( {\frac{{2\,x{e^x}}}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c$

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

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

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

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JEE • MATHEMATICS • INDEFINITE INTEGRATION

Indefinite integration practice · Medium

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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 301-330 of 411 questions

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301 Medium

301 of 411

Analyzing +4 | -1

The value of the integral

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

View answer and explanation

Correct option: C

\frac{{{e^x}}}{{1 + {x^2}}}\,\, + \,\,c

Option C is correct because the integral can be solved by rewriting the integrand and using integration by parts, leading to the result e^x / (1 + x²) + c.

302 Medium

302 of 411

Analyzing +4 | -1

f(x)

is a quadratic expression such that

f(0)=f(1)=3 f(2)=-3

, then

\int \frac{f(x)}{x^3-1} d x=

View answer and explanation

Correct option: A

\log _e|x-1|+\log _e\left(x^2+x+1\right)+\frac{2}{\sqrt{3}} \operatorname{Tan}^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)+c

The correct answer is A (

\log _e|x-1|+\log _e(x^2+x+1)+\frac{2}{\sqrt{3}} \operatorname{Tan}^{-1}(\frac{2 x+1}{\sqrt{3}})+c

) because after determining the quadratic expression

f(x)

, the integral is solved using partial fraction decomposition and standard integration techniques.

303 Medium

303 of 411

Analyzing +4 | -1

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

View answer and explanation

Correct option: C

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

Option B (

\frac{{x e^x}}{{\sqrt{{x^2+1}}}} + c

) is correct because the integral can be expressed in the form

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

, where

f(x) = \frac{{x}}{{\sqrt{{x^2+1}}}}

and its derivative is

\frac{{1}}{{(x^2+1)^{3/2}}}

304 Medium

304 of 411

Application +4 | -1

\int \nolimits_0^3 \,\,\frac{{3x\,\, + \,\,1}}{{{x^2}\,\, + \,\,9\;}}\;\,\,dx

is equal to

View answer and explanation

Correct option: A

\log \,\,(2\,\sqrt 2 )\,\, + \,\,\frac{\pi }{{12}}

Option A (

\log \,(2\sqrt 2 )\,+ \frac{\pi }{{12}}

) is correct because it results from correctly evaluating the definite integral after splitting it into two simpler integrals.

305 Medium

305 of 411

Analyzing +4 | -1

\int \,\,\frac{{{x^2}\,\, - \,\,1}}{{{x^4}\,\, + \,\,{x^2}\,\, + \,\,1}}\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}\,\,\log \,\,\frac{{{x^2}\,\, - \,\,x\,\, + \,\,1}}{{{x^2}\,\, + \,\,x\,\, + \,\,1}}\,\, + \,\,c

Option C (

\frac{1}{2}\,\,\log \,\,\frac{{{{x^2}}\,\, - \,\,x\,\, + \,\,1}}{{{{x^2}}\,\, + \,\,x\,\, + \,\,1}}\,\, + \,\,c

) is correct because the integral is solved by dividing by

x^2

and then using a substitution.

306 Medium

306 of 411

Application +4 | -1

\int \frac{x+\left(\cos ^{-1} 3 x\right)^2}{\sqrt{1-9 x^2}} d x=A \sqrt{1-9 x^2}+B\left(\cos ^{-1} 3 x\right)^3+c

, where

{c}

is integration constant, then the values of

{A}

and

{B}

are:

View answer and explanation

Correct option: A

{A}=-\frac{1}{9}, {~B}=-\frac{1}{9}

Option A (

{A}=-\frac{1}{9}, {~B}=-\frac{1}{9}

) is correct because it represents the correct values of A and B after integration and comparison.

307 Medium

307 of 411

Analyzing +4 | -1

I = \int \,\,\frac{1}{{{x^4}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} }}\,\,dx

, then I equals

View answer and explanation

Correct option: B

\frac{1}{{{a^4}}}\,\,\left\{ {\frac{1}{x}\,\,\sqrt {{a^2}\,\, + \,\,{x^2}} \,\, - \,\,\frac{1}{{3{x^3}}}\,\,{{\left( {{a^2}\,\, + \,\,{x^2}} \right)}^{3/2}}} \right\} + C

Option D (

\frac{1}{{{a^4}}}\,\left\{{ {\frac{1}{x}\,\sqrt {{a^2}\, + \,{x^2}} \, - \,{\frac{1}{{3{x^3}}}\,{{\left( {{a^2}\, + \,{x^2}} \right)}^{3/2}}} }\right\} + C

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

308 Medium

308 of 411

Application +4 | -1

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

, then

f(x)

is equal to

View answer and explanation

Correct option: C

\tan \frac{x}{2}

Option C (

\tan \frac{x}{2}

) is correct because it is the function that satisfies the given integral equation after simplification and integration.

309 Medium

309 of 411

Application +4 | -1

\int f\left( x \right)\,\,\sin \,\,x\,\,\cos \,\,x\,\,\;dx = \frac{1}{{2\,\,\left( {{b^2} - \,\,{a^2}} \right)}}\log \,\,\left\{ {f\left( x \right)} \right\} + C

, then

f\left( x \right)

is equal to

View answer and explanation

Correct option: A

\frac{1}{{{a^2}\,\,{{\sin }^2}x\,\, + \,\,{b^2}\,\,{{\cos }^2}\,\,x}}

Option A (

\frac{1}{{{a^2}\,{{\sin }^2}x\, + \,{b^2}\,{{\cos }^2}\,x}}

) is correct because by differentiating the given equation and solving for

f(x)

, we find that

f(x)

matches this form.

310 Medium

310 of 411

Application +4 | -1

\int \frac{2^{x+1}-5^{x-1}}{10^x} d x

is equal to

View answer and explanation

Correct option: B

\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c

Option B (

\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c

) is correct because it is the result of integrating the given expression using exponential integral formulas.

311 Medium

311 of 411

Application +4 | -1

\int \,\,\frac{{\sin \,\,x}}{{\sin \,(x\,\, - \,\,\alpha )}}\,\,dx

View answer and explanation

Correct option: D

x\,\,\cos \,\,\alpha \,\, + \,\,\sin \,\,\alpha \,\,\log \,\,\sin \,\,\left( {x - \alpha } \right)\,\, + \,\,c

Option D (

x\cos \alpha + \sin \alpha \log \sin (x - \alpha ) + c

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

312 Medium

312 of 411

Application +4 | -1

\int \nolimits_0^\lambda \,\,\frac{{y\,\,dy}}{{\sqrt {y + \lambda } }}

is equal to

View answer and explanation

Correct option: A

\frac{2}{3}\,\,\left( {2 - \sqrt 2 } \right)\,\,\lambda \,\,\sqrt \lambda

Option A is correct because after performing the substitution and evaluating the definite integral, the result matches the given expression.

313 Medium

313 of 411

Analyzing +4 | -1

\int x\,\,\log \,\,\left( {1\,\, + \,\,1/x} \right)\,\,dx = f\,\,\left( x \right)\,\, \cdot \,\,\log \,\,\left( {x\,\, + \,\,1} \right)\,\, + \,\,g\left( x \right)\,\, \cdot \,\,{x^2}\,\, + \,\,Ax + C

, then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given conditions do not match the derived functions f(x), g(x), and the value of A after integration and comparison.

314 Medium

314 of 411

Application +4 | -1

\int\left(x^6+x^4+x^2\right) \sqrt{2 x^4+3 x^2+6} d x

is equal to

View answer and explanation

Correct option: C

\frac{\left(2 x^6+3 x^4+6 x^2\right)^{3 / 2}}{18}+C

Option C (

\frac{\left(2 x^6+3 x^4+6 x^2\right)^{3 / 2}}{18}+C

) is correct because it results from a u-substitution where the derivative of the expression inside the square root is present in the integrand.

315 Medium

315 of 411

Application +4 | -1

\int \,\,{\cos ^4}\,x\,\,dx = Ax\,\, + \,\,B\,\,\sin \,\,2\,x\,\, + \,\,C\,\,\sin \,\,4x\,\, + \,\,D

, then {A,B,C} equals

View answer and explanation

Correct option: B

\left\{ {\frac{3}{8},\,\,\frac{1}{4},\,\,\frac{1}{{32}}} \right\}

Option B (

\left\{{{\frac{3}{8},\\,\frac{1}{4},\\,\frac{1}{{32}}}}\right\}

) is correct because the integral of

\cos^4 x

is evaluated using trigonometric identities and then the coefficients are matched.

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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 Medium

316 of 411

Application +4 | -1

Let

\mathrm{k}(\mathrm{x})=\int \frac{\left(x^2+1\right) d x}{\sqrt[3]{x^3+3 x+6}}

and

\mathrm{k}(-1)=\frac{1}{\sqrt[3]{2}}

then the value of

\mathrm{k}(-2)

View answer and explanation

Correct option: A

Option A (2) is correct because after integration and using the given condition k(-1), we find the constant C=0, and then k(-2) = 2.

317 Medium

317 of 411

Application +4 | -1

\int \frac{e \cos (e \ln x) d x}{x}

is equal to

View answer and explanation

Correct option: B

\sin (e \ln x)+c

Option B (

\sin (e \ln x)+c

) is correct because the integral is solved using substitution, where

e \ln x = t

318 Medium

318 of 411

Analyzing +4 | -1

\int \,{\left( {\frac{{x + 2}}{{x + 4}}} \right)^2}{e^x}\,dx

is equal to

View answer and explanation

Correct option: A

{e^x}\,\,\left( {\frac{x}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c

Option A is correct because the integral simplifies to

{e^x}\,\left( {\frac{x}{{x\,\, + \,\,4}}} \right)\,\, + \,\,c

after applying integration by parts and simplification.

319 Medium

319 of 411

Analyzing +4 | -1

Let

f

be a polynomial function such that for all real

x, f\left(x^2+1\right)=x^4+5 x^2+2

, then

\int f(x) d x

View answer and explanation

Correct option: A

\frac{x^3}{3}+\frac{3 x^2}{2}-2 x+c

Option A,

\frac{x^3}{3}+\frac{3 x^2}{2}-2 x+c

, is correct because by substituting

x^2+1

into f, we can determine f(x) and then integrate it.

320 Medium

320 of 411

Analyzing +4 | -1

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

View answer and explanation

Correct option: C

-\operatorname{Sec}^{-1}(\sin 2 x+1)+c

The correct answer is C (

-\operatorname{Sec}^{-1}(\sin 2 x+1)+c

) because the integral simplifies to a standard inverse secant form after appropriate trigonometric manipulations and substitution.

321 Medium

321 of 411

Application +4 | -1

\int \frac{a \sin x+b \cos x}{c \sin x+d \cos x} d x=p x+q \log _e|c \sin x+d \cos x|+c

where

p+q

is equal to

View answer and explanation

Correct option: A

\frac{c(a+b)-d(a-b)}{c^2+d^2}

Option A (

\frac{{c(a+b)-d(a-b)}}{{c^2+d^2}}

) is correct because it represents the value of

p+q

obtained by decomposing the integrand and integrating.

322 Medium

322 of 411

Application +4 | -1

\int {{x^6}\sin \left( {5{x^7}} \right)dx = \frac{k}{5}\cos \left( {5{x^7}} \right),x} \ne 0,

then

View answer and explanation

Correct option: D

k = - \frac{1}{7}

Option D (

k = - \frac{1}{7}

) is correct because after performing u-substitution, the integral evaluates to

-\frac{1}{35}\cos(5x^7)

, which implies

k = -\frac{1}{7}

323 Medium

323 of 411

Application +4 | -1

F_n(x)=\int(\log x)^n d x

and

\phi_n=F_n+n F_{n-1}

, then

\phi_n(e)-\phi_n(1)

is equal to

View answer and explanation

Correct option: D

e

Option D is correct because by using integration by parts, we find that φ_n(x) = x(log x)^n + c, and then φ_n(e) - φ_n(1) = e.

324 Medium

324 of 411

Application +4 | -1

Let

f

be a real-valued function defined on the interval

\left( { - 1,\,1} \right)

such that

{e^{ - x}}\,\,f\,\,\left( x \right) = 2\,\, + \,\,\int \nolimits_0^x \,\,\sqrt {{t^4}\,\, + \,\,1} \,\,dt

, for all

x \in \left( { - 1,\,\,1} \right)

and let

{f^{ - 1}}

be the inverse function of

f

. Then,

{\left( {{f^{ - 1}}} \right)^\prime }\,\,\left( 2 \right)

is equal to

View answer and explanation

Correct option: B

\frac{1}{3}

Option B (1/3) is correct because after differentiating and using the inverse function property, the derivative of the inverse function at 2 is found to be 1/3.

325 Medium

325 of 411

Application +4 | -1

Let

f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x(x \geq 0)

. Then

f(3)-f(1)

is equal to :

View answer and explanation

Correct option: D

\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}

Option D

\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}

is correct because it is the result of evaluating the definite integral after appropriate substitution and simplification.

326 Medium

326 of 411

Application +4 | -1

\int \frac{\sin x}{\sin x-\cos x} d x

equals

View answer and explanation

Correct option: A

\frac{1}{2} \log _e|\sin x-\cos x|+\frac{x}{2}+c

Option A is correct because the integral is solved by rewriting the numerator and separating the integral into simpler parts, resulting in (1/2)log_e|sinx-cosx| + x/2 + c.

327 Medium

327 of 411

Analyzing +4 | -1

\int(\sin 4 x) e^{\tan ^2 x} d x=

View answer and explanation

Correct option: A

-2 e^{\tan ^2 x} \cos ^4 x+c

Option A (

-2 e^{\tan ^2 x} \cos ^4 x+c

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

328 Medium

328 of 411

Understanding +4 | -1

The relation

\int(\sin 2 x+\cos 2 x) d x=\frac{1}{\sqrt{2}} \sin (2 x-a)+c

is true for

a

and

c

equal to -

View answer and explanation

Correct option: C

a=\pi / 4, c

is arbitrary constant

Option C (

a=\pi / 4, c

is arbitrary constant) is correct because the integral of the given expression results in a sine function with a phase shift of

\pi/4

and an arbitrary constant of integration.

329 Medium

329 of 411

Application +4 | -1

\int \frac{\left(x-x^5\right)^{1 / 5}}{x^6} d x

is equal to

View answer and explanation

Correct option: C

-\frac{5}{24}\left(\frac{1}{x^4}-1\right)^{6 / 5}+C

Option C is correct because the integral is solved using a substitution method, resulting in the given expression with a fractional exponent.

330 Medium

330 of 411

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the integral simplifies to this expression after 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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