If $f(x)=f(a+b-x)$, then $\int_\limits a^b x f(x) d x$ is equal to

$(b-a) \int_\limits a^b f(x) d x$

$\frac{1}{2}(b-a) \int_\limits a^b f(x) d x$

The value of the integral $\mathop \smallint \limits_{ - \pi /3}^{\pi /3} \,\,\frac{{x\,\,\sin \,\,x}}{{{{\cos }^2}\,\,x}}\,\,dx$, is

$\left( {\pi /3\,\, - \,\,\log \,\,\tan \,\,3\pi /2} \right)$

$3\,\,\left( {\pi /2\,\, - \,\,\log \,\,\sin \,\,\pi /12} \right)$

The value of $\int_\limits 0^4 3^{\sqrt{2 x+1}} \mathrm{~d} x$ is

$\frac{6}{\log 3}\left(13-\frac{4}{\log 3}\right)$

$\frac{6}{\log 3}\left(13-\frac{5}{\log 3}\right)$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite 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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Showing 391-420 of 995 questions

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391 Easy

391 of 995

Application +4 | -1

f(x)=\int_0^{\sin x} \cos ^{-1} t d t+\int_0^{\cos x} \sin ^{-1} t d t

0 < x < \frac{\pi }{2}

, then

f(\pi / 4)

is equal to

View answer and explanation

Correct option: B

1+\frac{\pi}{2 \sqrt{2}}

Option C (

-\frac{1}{2}

) is correct because it is the value of the function f(x) after differentiating the given integral equation and substituting

x = \frac{\pi}{6}

392 Easy

392 of 995

Application +4 | -1

The value of the integral ∫

\int \limits_0^{\pi /4} \,\,\frac{{\sin \,\,\theta \,\, + \cos \,\,\theta }}{{9\,\, + \,\,16\,\,\sin \,\,2\theta }}\,\,d\theta

, is

View answer and explanation

Correct option: A

\frac{3}{8}

Option A (

\frac{3}{8}

) is correct because it is the value obtained after evaluating the definite integral using trigonometric substitution.

393 Medium

393 of 995

Application +4 | -1

The value of

\int \nolimits_{ - 1}^3 \,\,\left[ {{{\tan }^{ - 1}}\,\,\left( {\frac{x}{{{x^2} + \,\,1}}} \right)\,\, + \,\,{{\tan }^{ - 1}}\,\,\left( {\frac{{{x^2} + 1}}{x}} \right)} \right]\;\,\,dx

View answer and explanation

Correct option: A

2\,\mathrm{\pi }

Option A (

2\pi

) is correct because the expression inside the integral simplifies to

\pi/2

, and the integral of a constant is straightforward.

394 Easy

394 of 995

Analyzing +4 | -1

\int_0^2\left(\sqrt{2 x}-\sqrt{2 x-x^2}\right) d x=\int_0^1\left(1-\sqrt{1-y^2}-\frac{y^2}{2}\right) d y+

\int_1^2\left(2-\frac{y^2}{2}\right) d y+I

then

I

equal is

View answer and explanation

Correct option: C

\int_0^1\left(1-\sqrt{1-y^2}\right) d y

Option C (

\int_0^1\left(1-\sqrt{1-y^2}\right) d y

) is the correct answer because it results from simplifying and rearranging the given integral equation.

395 Medium

395 of 995

Application +4 | -1

The value of the integral

\int \limits_0^{\pi /2} \,\,\left| {\,\sin \,\,x\,\, - \,\,\cos \,\,x\,} \right|\,\,dx

, is

View answer and explanation

Correct option: B

2\,\,\left( {\sqrt 2 - 1} \right)

Option B (

2\,\,\left( {\sqrt 2 - 1} \right)

) is correct because the integral of the absolute value function splits into two integrals, and the result is

2\sqrt{2} - 2

396 Easy

396 of 995

Application +4 | -1

f(x)

is differentiable and

\int\limits_{0}^{t^2} x f(x) d x=\frac{2}{5} t^5

then

f(4 / 25)=

View answer and explanation

Correct option: A

\frac{2}{5}

Option A (

\frac{2}{5}

) is correct because differentiating both sides and substituting

t = \frac{2}{5}

gives

f(\frac{4}{25}) = \frac{2}{5}

397 Easy

397 of 995

Application +4 | -1

f\,\,\left( x \right) = \,\,\left\{ {\begin{array}{cccccccccccccccccccc} {1 - x,}\\ {0,}\\ {{{\left( {2 - x} \right)}^2},} \end{array}\,\,\begin{array}{cccccccccccccccccccc} {0 \le \,\,\,x\,\, \le 1}\\ {1 \le \,\,\,x\,\, \le 2}\\ {2 \le \,\,\,x\,\, \le 3} \end{array}} \right.

and

\phi \,\,\left( x \right) = \int \limits_0^x \,\,f\,\left( t \right)\,\,dt

. Then, for any

x \in \left[ {2,3} \right],\,\,\phi \,\,\left( x \right)

equals

View answer and explanation

Correct option: C

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

Option C is correct because the integral of f(t) from 0 to x is calculated piecewise, resulting in the given expression for x in [2,3].

398 Hard

398 of 995

Analyzing +4 | -1

\int \limits_{ - \pi /2}^{\pi /2} \,\,{\log _e}\,\,\left\{ {\left( {\frac{{a{x^2} + \,\,bx + c}}{{a{x^2} - \,\,bx + c}}} \right)\,\,\left( {a + b} \right)\,\,\left| {\,\sin \,\,x\,} \right|} \right\}\,\,dx

is equal to

View answer and explanation

Correct option: B

\pi \,\,{\log _e}\,\,\left( {\frac{{a + b}}{2}} \right)

Option B (

\pi \,{\log _e}\,\left( {\frac{{a + b}}{2}} \right)

) is correct because the integral simplifies using properties of logarithms and even/odd functions.

399 Medium

399 of 995

Application +4 | -1

f(x)=A \sin \frac{\pi x}{2}+B, f^{\prime}(1 / 2)=\sqrt{2}

and

\int\limits_{0}^1 f(x) \mathrm{d} x=\frac{2 A}{\pi}

then the constants

A

and

B

are respectively

View answer and explanation

Correct option: D

4 / \pi, 0

Option D (4/

\pi

, 0) is correct because solving the given equations for A and B yields these values.

400 Easy

400 of 995

Application +4 | -1

\int\limits_{0}^2\left[x^2\right] d x

is equal to

View answer and explanation

Correct option: B

5-\sqrt{3}-\sqrt{2}

Option B (

5-\sqrt{3}-\sqrt{2}

) is correct because the integral of the greatest integer function of x squared from 0 to 2 is equal to

5-\sqrt{3}-\sqrt{2}

401 Medium

401 of 995

Application +4 | -1

\int \nolimits_{\log 2}^x \,\,\frac{{du}}{{{{\left( {{e^u} - 1} \right)}^{1/2}}}} = \frac{\pi }{6}

, then

{e^x}

is equal to

View answer and explanation

Correct option: C

\mathrm{4}

Option C (4) is correct because it is the result of solving the definite integral and then solving for e^x.

402 Medium

402 of 995

Analyzing +4 | -1

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

is equal to

View answer and explanation

Correct option: C

- \,\,\frac{\pi }{2}\,\,\log \,\,2

Option C

\left( { - \,\,\frac{\pi }{{2}}\,\log \,2} \right)

is correct because by using the property

\int \nolimits_0^{a} f(x) dx = \int \nolimits_0^{a} f(a-x) dx

and trigonometric identities, the integral evaluates to

-\frac{\pi}{2}\log 2

403 Easy

403 of 995

Application +4 | -1

f(x)=f(a+b-x)

, then

\int\limits_{a}^b x f(x) d x

is equal to

View answer and explanation

Correct option: B

\frac{1}{2}(a+b) \int\limits_{a}^b f(x) d x

Option B is correct because using the property of definite integrals

\int_a^b f(x) dx = \int_a^b f(a+b-x) dx

, we can simplify the given integral and find its value.

404 Medium

404 of 995

Application +4 | -1

The value of

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

(where

[

] denotes greatest

integer function) is

View answer and explanation

Correct option: B

Option B (1) is correct because the integral evaluates to 1 after splitting the integral and using the properties of the greatest integer function.

405 Easy

405 of 995

Application +4 | -1

If, for

t>0

the definite integral

\int\limits_{0}^{t^2} x f(x) d x=\frac{2}{5} t^5

, then

f\left(\frac{4}{25}\right)

is equal to

View answer and explanation

Correct option: A

\frac{2}{5}

Option A (

\frac{2}{5}

) is correct because differentiating the given integral and solving for

f(t^2)

yields

f(\frac{4}{25}) = \frac{2}{5}

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

406 Medium

406 of 995

Application +4 | -1

If the integral

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

where

k

is an arbitrary constant, then

A

is equal to:

View answer and explanation

Correct option: A

-\frac{1}{16}

Option A (-1/16) is correct because the integral simplifies to a cosine function with a coefficient of -1/16.

407 Medium

407 of 995

Application +4 | -1

If the

\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k

, then

a

is equal to:

View answer and explanation

Correct option: D

Option D (2) is correct because after simplifying the integral, the coefficient of the natural logarithm term is 2.

408 Medium

408 of 995

Application +4 | -1

The value of the integral

\int \limits_{ - \pi /3}^{\pi /3} \,\,\frac{{x\,\,\sin \,\,x}}{{{{\cos }^2}\,\,x}}\,\,dx

, is

View answer and explanation

Correct option: B

2\,\,\left( {2\pi /3\,\, - \,\,\log \,\,\tan \,\,5\pi /12} \right)

Option B (

2\,\left( {2\pi /3\, - \,\log \,\tan \,5\pi /12} \right)

) is correct because it is the result of evaluating the definite integral using integration by parts and properties of even functions.

409 Easy

409 of 995

Analyzing +4 | -1

The value of the integral

\int\limits_1^3 {\left[ {{x^2} - 2x - 2} \right]\,\,\,dx} \,

where

[x]

denotes the greatest integer less than or equal to

x

, is

View answer and explanation

Correct option: B

- \sqrt 2 - \sqrt 3 - 1

Option B (

\ - \sqrt 2 - \sqrt 3 - 1

) is correct because evaluating the definite integral of the greatest integer function results in this value.

410 Easy

410 of 995

Application +4 | -1

\int\limits_{0}^\infty \frac{\sin x}{x} d x=\frac{\pi}{2}

then

\int\limits_{0}^\infty \frac{\sin ^3 x}{x} d x=

View answer and explanation

Correct option: B

\frac{\pi}{4}

Option B (

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

) is correct because it is the result of using the trigonometric identity to express

\sin^3 x

and then applying the given integral value.

411 Medium

411 of 995

Application +4 | -1

\int_0^{16 n^2 / \pi} \cos \frac{\pi}{2}\left[\frac{x \pi}{n}\right] d x

is equal to

View answer and explanation

Correct option: A

Option A is correct because by substituting and using the properties of the greatest integer function and cosine, the integral evaluates to 0.

412 Medium

412 of 995

Application +4 | -1

f\left( \pi \right) = 2

and

\int \nolimits_0^\pi \,\,\left[ {f\left( x \right)\,\, + \,\,f''\left( x \right)} \right]\,\,\sin \,\,x\,\,dx = 5,

, then

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

is equal to, (it is given that

f\left( x \right)

is continuous in

\left[ {0,\,\,\pi } \right]\,

)

View answer and explanation

Correct option: B

Option B (3) is correct because integration by parts is applied to simplify the integral, leading to the equation

f(\pi) + f(0) = 5

, and given

f(\pi) = 2

, we find

f(0) = 3

413 Easy

413 of 995

Analyzing +4 | -1

I_n=\int\limits_{{\pi / 4}}^{\pi / 2} \cot ^n x \,d x

, then

I_1+I_3, I_2+I_4, I_3+I_5, I_4+I_6, \ldots

are in

View answer and explanation

Correct option: C

H.P.

Option C (H.P.) is correct because the reciprocals of the given terms form an arithmetic progression.

414 Medium

414 of 995

Application +4 | -1

The value of

\int \limits_0^1 \left( {1\,\, + \,\,{e^{ - {x^2}}}} \right)dx

, is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the definite integral's value falls within a range that doesn't match any of the provided options. The integral cannot be expressed in a simple closed form.

415 Easy

415 of 995

Analyzing +4 | -1

Let

{J_{n,\,\,m}} = \int\limits_0^{\frac{1}{2}} {\frac{{{x^n}}}{{{x^m} - 1}}} dx,\,\forall n > m\mathrm{ }\,\,\mathrm{and}\,\,n,\,\,m \in N.

Consider a matrix

A = {[{a_{ij}}]_{3 \times 3}}

where

{a_{ij}} = \left\{ {\begin{array}{cccccccccccccccccccc} {{J_{6 + i,3}} - {J_{i + 3,3}}}&{i \le j}\\ {0,}&{i > j} \end{array}} \right.

Then,

|\mathrm{adj }\,\,{\mathrm{A}^{ - 1}}|

View answer and explanation

Correct option: C

{(105)^2} \times {2^{38}}

The correct answer is D (

{(105)^2} \times {2^{36}}

) after evaluating the determinant of the adjoint of the inverse of matrix A, which involves integral calculations and properties of determinants.

416 Medium

416 of 995

Application +4 | -1

The value of

\int\limits_{0}^{\pi / 2} \frac{\sin ^3 x}{\sin x+\cos x} d x

is:

View answer and explanation

Correct option: B

\frac{\pi-1}{4}

Option B (

\frac{{\pi-1}}{{4}}

) is correct because it is the result obtained after applying the property of definite integrals and simplifying the expression.

417 Medium

417 of 995

Application +4 | -1

The value of

\alpha

for which

4 \alpha \int\limits_{{-1}}^2 e^{-\alpha|x|} d x=5

, is :

View answer and explanation

Correct option: A

\log _e 2

Option A (

\log_e 2

) is correct because it is the value of

\alpha

that satisfies the given integral equation after solving for

\alpha

418 Medium

418 of 995

Application +4 | -1

The value of

\int\limits_{0}^4 3^{\sqrt{2 x+1}} \mathrm{~d} x

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to a value not present in the given options.

419 Easy

419 of 995

Application +4 | -1

\int \nolimits_{ - 2}^2 \,\,\left| {\,x\,} \right|\;\,\,dx\;

is equal to

View answer and explanation

Correct option: D

\mathrm{4}

Option D (

\mathrm{4}

) is correct because the integral of the absolute value function is calculated by splitting the integral at the point where the function changes sign.

420 Easy

420 of 995

Application +4 | -1

\int\limits_{{\pi / 2}}^\theta \sin x \, d x=\sin 2 \theta

, then the value of

\theta

satisfying

0 < \theta < \pi

, is

View answer and explanation

Correct option: D

\pi / 2

Option D (

\frac{\pi}{2}

) is correct because solving the integral and the resulting trigonometric equation leads to

\theta = \frac{\pi}{2}

within the given range.

Revision notes and key takeaways

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

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