The value of $\mathop \smallint \nolimits_0^{\pi /2} \,\,{\sin ^8}\,\,x\,dx$ is

$\frac{{105\,\pi }}{{14\left( {4\,!} \right)}}$

$\frac{{105}}{{16\,\pi \left( {4!} \right)}}$

$\mathop \smallint \nolimits_{ - 100}^{100} \,\,f\,\,\left( x \right)\,\,dx$ is equal to

$\mathop \smallint \nolimits_{ - 100}^{100} \,\,f\left( { - x} \right)\,\,dx$

$\mathop \smallint \nolimits_{ - 100}^{100} \,\,f\left( {{x^2}} \right)\,\,dx$

$\mathop \smallint \nolimits_{ - 100}^{100} \,\,f\left( { - \,{x^2}} \right)\,\,dx$

$\mathop \smallint \nolimits_{ - 100}^{100} \,\,f\left( {\frac{1}{x}} \right)\,\,dx$

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

$\log \,\,\left( {2\,\,\sqrt 2 } \right)\,\, + \,\,\frac{{\rm{\pi }}}{{12}}$

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

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

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

Back to DEFINITE INTEGRATION

Your Progress

0 / 995 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

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.

Take a chapter quiz →

Showing 301-330 of 995 questions

Preparing practice and saved progress…

301 Medium

301 of 995

Application +4 | -1

\int\limits_{0}^{102}\left[\tan ^{-1} x\right] {dx}=(

where

[x]

is the largest integer not exceeding

x)

View answer and explanation

Correct option: A

102-\tan 1

Option A (

102-\tan 1

) is correct because the integral can be split into two parts based on the value of the greatest integer function, and then evaluated.

302 Easy

302 of 995

Application +4 | -1

\int \nolimits_2^k \,\,\left( {2x + 1} \right)\,\,dx = 6

, then

k

is equal to

View answer and explanation

Correct option: D

\mathrm{3}

Option A (

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

) is correct because the integral can be solved by rewriting the denominator and using trigonometric identities.

303 Easy

303 of 995

Application +4 | -1

\int \nolimits_{ - 1}^0 \,\,\frac{{dx}}{{{x^2} + 2x + 2}}

is equal to

View answer and explanation

Correct option: B

\mathrm{\pi }/4

Option B (

\frac{\mathrm{\pi }}{4}

) is correct because the definite integral of

\frac{{dx}}{{{x^2} + 2x + 2}}

from -1 to 0 is

\frac{\pi}{4}

304 Easy

304 of 995

Analyzing +4 | -1

The value of the integral

\int \limits_0^\pi \,\,\frac{{\sin \,\,k\,\;x}}{{\sin \,\,x}}\,\,dx

is an even integer) is equal to

View answer and explanation

Correct option: D

\mathrm{0}

Option D (

\mathrm{0}

) is correct because the integral evaluates to 0 due to the symmetry of the sine function.

305 Easy

305 of 995

Application +4 | -1

The value of

\int \nolimits_0^1 \,\,\frac{{dx}}{{x\,\, + \,\,\sqrt {1\,\, - \,\,{x^2}} }}

View answer and explanation

Correct option: D

\frac{\mathrm{\pi }}{4}

Option D (

\frac{\mathrm{\pi }}{4}

) is correct because the integral is solved using a trigonometric substitution, and the result is

\frac{\pi}{4}

306 Easy

306 of 995

Application +4 | -1

The value of

\int \limits_{ - 1}^3 \,\,\left( {\,\left| {x - 2} \right| + \,\left[ x \right]\,} \right)\,\,dx

is (

\left[ x \right]

stands for greatest integer less than or equal to

x

)

View answer and explanation

Correct option: A

\mathrm{7}

Option A (7) is correct because it is the result of evaluating the definite integral by splitting the integral at the points where the absolute value and greatest integer functions change their behavior.

307 Easy

307 of 995

Application +4 | -1

f\left( x \right) = \int \nolimits_{ - 1}^x \,\,\left| {\,t\,} \right|\,\,dt

, then for any

x \ge 0,\,\,f\left( x \right)

is equal to

View answer and explanation

Correct option: B

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

Option B is correct because the integral of the absolute value function is split into two parts and then evaluated.

308 Easy

308 of 995

Application +4 | -1

The value of

\int \nolimits_0^{\pi /2} \,\,{\sin ^8}\,\,x\,dx

View answer and explanation

Correct option: A

\frac{{105\,\,\pi }}{{32\,\,\left( {4\,\,!} \right)}}

Option A is correct because the integral is evaluated using Walli's formula, resulting in the given expression.

309 Easy

309 of 995

Application +4 | -1

The value of

\int \nolimits_0^1 \,\,\log \,\,\left[ x \right]\,\,dx

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the logarithm of 0 is undefined, making the integral undefined.

310 Medium

310 of 995

Application +4 | -1

\lim\limits_{x \to \infty } \,\,\frac{\pi }{n}\,\,\left\{ {\sin \,\,\frac{\pi }{n}\,\, + \,\,\sin \,\,\frac{{2\,\pi }}{n}\,\, + \ldots + \,\,\sin \,\,\frac{{\left( {n - 1} \right)\pi }}{n}} \right\}

equals

View answer and explanation

Correct option: C

Option C (2) is correct because the limit of the Riemann sum can be expressed as a definite integral, which evaluates to 2.

311 Easy

311 of 995

Analyzing +4 | -1

f(x) = \frac{{2 - x\,\cos \,x}}{{2 + \,x\,\cos \,x}}

and

g(x) = {\log _e}\,x,\,\,(x\, > \,0)

then the value of the integral

\int_{ - \pi /4}^{\pi /4} \, g(f(x))dx

View answer and explanation

Correct option: D

{\log _e}1

Option D (

{\log _e}1

) is correct because the integral evaluates to 0, which is the natural logarithm of 1.

312 Easy

312 of 995

Application +4 | -1

Let

f:(0, \infty) \rightarrow \mathbf{R}

and let

F(x)=\int\limits_{0}^x f(t) d t

F\left(x^2\right)=x^2(1+x)

, then

f(4)

equals

View answer and explanation

Correct option: C

Option C (4) is correct because by using the fundamental theorem of calculus and differentiating the given equation, we can find f(4).

313 Easy

313 of 995

Application +4 | -1

g(x)=\int\limits_{0}^x \cos ^4 t \mathrm{~d} t

then

g(x+\pi)

equals

View answer and explanation

Correct option: A

g(x)+g(\pi)

Option A (g(x)+g(π)) is correct because the integral from 0 to x+π can be split into integrals from 0 to π and π to x+π, and a substitution can be used to show the latter equals g(x).

314 Easy

314 of 995

Application +4 | -1

n \in N

, then

\int\limits_{{-n}}^n(-1)^{[x]} d x

equals

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.

315 Easy

315 of 995

Application +4 | -1

The value of

\int\limits_{{-1}}^3\{|x-2|+[x]\} d x

, where

[x]

denotes the greatest integer less than or equal to

x

View answer and explanation

Correct option: B

Option B (7) is correct because the integral is evaluated by splitting it into intervals based on the absolute value and greatest integer functions.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs DIFFERENTIAL EQUATIONS 977 MCQs

316 Easy

316 of 995

Application +4 | -1

The value of

\int\limits_{{-\pi / 2}}^{\pi / 2} \frac{\sin ^2 x}{1+2^x} d x

is :

View answer and explanation

Correct option: C

\frac{\pi}{4}

Option C (

\frac{\pi}{4}

) is correct because the integral is solved using the property of definite integrals and trigonometric identities.

317 Medium

317 of 995

Analyzing +4 | -1

For any integer

n

, the integral

\int \nolimits_0^\pi \,\,{e^{{{\cos }^2}x}}\,\,{\cos ^3}\left( {2n + 1} \right)\,\,x\,\,\;dx\;

has the value

View answer and explanation

Correct option: C

\mathrm{0}

Option C (0) is correct because the integrand is an odd function with respect to x = π/2, making the integral over the interval [0, π] equal to zero.

318 Easy

318 of 995

Understanding +4 | -1

\int \nolimits_{ - 100}^{100} \,\,f\,\,\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

\int \nolimits_{ - 100}^{100} \,\,f\left( { - x} \right)\,\,dx

Option D (

\int \nolimits_{ - 100}^{100} \,f\left( { - x} \right)\,dx

) is correct because substituting

x = -t

results in the same integral.

319 Easy

319 of 995

Application +4 | -1

\int_0^{[x] / 3} \frac{8^x}{2^{[3 x]}} d x

where

[\cdot]

denotes the greatest integer function, is equal to

View answer and explanation

Correct option: D

\frac{[x]}{\ln 8}

The correct answer is D (

\frac{[x]}{\ln 8}

) because after simplifying the integrand and using the properties of the greatest integer function, the integral evaluates to

\frac{[x]}{\ln 8}

320 Easy

320 of 995

Understanding +4 | -1

\int\limits_{{-1}}^1 x^{17} \cos ^4 x d x=

View answer and explanation

Correct option: C

Option C (0) is correct because the integrand is an odd function, and the integral of an odd function over a symmetric interval is zero.

321 Easy

321 of 995

Application +4 | -1

The correct evaluation of

\int\limits_{0}^{\pi / 2}\left|\sin \left(x-\frac{\pi}{4}\right)\right| d x

View answer and explanation

Correct option: B

2-\sqrt{2}

Option B (

2-\sqrt{{2}}

) is correct because it is the result of evaluating the definite integral of the absolute value of a sine function over the given interval.

322 Easy

322 of 995

Analyzing +4 | -1

Given

\int\limits_{1}^2 e^{x^2} d x=a

, the value of

\int\limits_{e}^{e^4} \sqrt{\log _e x} \,d x

, is

View answer and explanation

Correct option: D

2 e^4-e-a

Option D (

2 e^4-e-a

) is correct because it is obtained by using substitution and integration by parts on the given integral.

323 Easy

323 of 995

Application +4 | -1

\int\limits_{{-1}}^1 \log \frac{2-x}{2+x} d x=

View answer and explanation

Correct option: D

Option D (0) is correct because the integrand is an odd function, and the integral is evaluated over a symmetric interval [-1, 1].

324 Medium

324 of 995

Application +4 | -1

The number of integral solutions of the equation

4 \int_0^{\infty} \frac{\ln t d t}{x^2+t^2}-\pi \ln 2=0 ; x>0

, is

View answer and explanation

Correct option: C

Option C is correct because solving the integral equation leads to two integral solutions for x, which are 2 and 4.

325 Easy

325 of 995

Application +4 | -1

\int \nolimits_0^\infty \,\,\left[ {\frac{{\left[ {x\,\, + \,\,1} \right]}}{{{e^x}}}} \right]\,\,dx

(where

\left[ {\,\, \cdot \,\,} \right]

denotes greatest integer function) is equal to

View answer and explanation

Correct option: A

\mathrm{0}

Option A (0) is correct because the expression inside the greatest integer function is always less than 1, making the greatest integer function equal to 0.

326 Medium

326 of 995

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 \,\,\left( {2\,\,\sqrt 2 } \right)\,\, + \,\,\frac{\mathrm{\pi }}{{12}}

Option A (

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

) is correct because it is the result of correctly splitting the integral and evaluating each part.

327 Medium

327 of 995

Application +4 | -1

The value of the integral

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

, is

View answer and explanation

Correct option: A

Option A (0) is correct because the integrand is an odd function and the limits of integration are symmetric about the origin.

328 Medium

328 of 995

Application +4 | -1

The value of the integral

\int \limits_{ - \pi }^\pi {\left( {\cos \,\,ax\,\, - \,\,\sin \,\,bx} \right)^2}dx

, where (

a

and

b

integers), is

View answer and explanation

Correct option: D

2\,\pi

Option D (

2\,\pi

) is correct because it involves integrating trigonometric functions over a symmetric interval, utilizing their properties.

329 Medium

329 of 995

Application +4 | -1

\int_0^{\pi / 3} \frac{\cos x+\sin x}{\sqrt{1+\sin 2 x}} d x

View answer and explanation

Correct option: D

\frac{\pi}{3}

Option D (

\frac{{\pi}}{{3}}

) is correct because the integral simplifies to

\int_0^{\pi / 3} dx

, which evaluates to

\frac{{\pi}}{{3}}

330 Easy

330 of 995

Analyzing +4 | -1

The value of

\int\limits_{{-1}}^1 \{\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\}\ d x

, is

View answer and explanation

Correct option: A

Option A (0) is correct because the integrand is an odd function integrated over a symmetric interval.

Revision notes and key takeaways

Review DEFINITE INTEGRATION in your textbook or class notes, then use this set to check your understanding.

Showing 301-330 of 995 questions

Your Progress

Definite integration practice · Mixed difficulty

Explore Other Chapters in MATHEMATICS

Continue Your Practice

Your practice