Let $I = \mathop \smallint \nolimits_0^1 \,\,\frac{{\sin \,\,x}}{{\sqrt x }}\,\,dx\,\,{\rm{and}}\,\,J = \mathop \smallint \nolimits_0^1 \,\,\frac{{\cos \,\,x}}{{\sqrt x }}\,\,dx$. Then, which one of the following is true?

$I < \frac{2}{3}\,\,{\rm{and}}\,\,J < 2$

$I\,\, > \,\,\frac{2}{3}\,\,{\rm{and}}\,\,J < \,\,2$

$I > \frac{2}{3}\,\,{\rm{and}}\,\,J > 2$

$I\,\,\left\langle {\,\frac{2}{3}\,\,{\rm{and}}\,\,J\,} \right\rangle \,\,2$

The integral $\mathop \smallint \limits_0^{\pi /2} \,\,f\,\,\left( {\sin \,\,2x} \right)\,\,\sin \,\,x\,\,dx$ is equal to

$\mathop \smallint \limits_0^{\pi /2} \,\,f\left( {\sin \,\,2x} \right)\,\,\cos \,\,x\,\,dx = \sqrt 2 \,\,\mathop \smallint \limits_0^{\pi /4} \,\,f\left( {\cos \,\,\,2x} \right)\cos \,\,x\,\,dx$

$\mathop \smallint \limits_0^{\pi /2} \,\,f\,\,\left( {\cos \,\,2x} \right)\,\,\sin \,\,x\,\,dx = \sqrt 2 \,\mathop \smallint \limits_0^{\pi /2} \,\,f\,\,\left( {\cos \,\,\,2x} \right)\sin \,\,x\,\,dx$

$\mathop \smallint \limits_0^{\pi /2} \,\,f\left( {\cos \,\,2x} \right)\cos \,\,x\,\,dx = \sqrt 2 \,\,\mathop \smallint \limits_0^{\pi /2} \,\,f\left( {\cos \,\,2x} \right)\,\,\cos \,\,x\,\,dx$

$\mathop \smallint \limits_0^{0\pi /2} \,\,f\left( {\sin \,\,2x} \right)\,\,\cos \,\,x\,\,dx = \sqrt 2 \,\,\mathop \smallint \limits_0^{\pi /2} \,\,f\left( {\cos \,\,\,2x} \right)\,\,\cos \,\,x\,\,dx$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite 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 791

Application +4 | -1

The integral

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

equals

View answer and explanation

Correct option: D

e(4 e-1)

Option D is correct. The integral is solved by recognizing the derivative of (ex)^x and evaluating the definite integral.

512 Easy

512 of 791

Understanding +4 | -1

The value of

\int\limits_{{-\pi / 2}}^{\pi / 2} \log \left(\frac{2-\sin \theta}{2+\sin \theta}\right) \mathrm{d} \theta

View answer and explanation

Correct option: A

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

513 Easy

513 of 791

Application +4 | -1

\int\limits_{n}^{n+1} f(x) d x=\frac{n^2}{2}

, where

n \in I

, then evaluate

\int\limits_{{-3}}^5 f(|x|) d x

View answer and explanation

Correct option: A

\frac{35}{2}

Option A (35/2) is correct because it is calculated by splitting the integral and using the given property

\int_n^{n+1} f(x) dx = \frac{{n^2}}{{2}}

514 Easy

514 of 791

Application +4 | -1

Area bounded by

y=5 x^{2}

y-9=2 x^{2}

View answer and explanation

Correct option: D

12 \sqrt{3}

Option D is correct because the area bounded by the two parabolas is found by integrating the difference of the functions between their intersection points.

515 Easy

515 of 791

Analyzing +4 | -1

I(m, n)=\int\limits_{0}^1 t^m(1-t)^n d t

, then the expression for

I(m, n)

in terms of

I(m+1, n-1)

, is

View answer and explanation

Correct option: B

\frac{n}{m+1} I(m+1, n-1)

The correct answer is B (

\frac{n}{m+1} I(m+1, n-1)

) because integration by parts allows expressing

I(m, n)

in terms of

I(m+1, n-1)

516 Easy

516 of 791

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: C

\mathrm{0}

Option C,

\mathrm{0}

, is correct because using the property of definite integrals and adding the original integral to the transformed integral results in zero.

517 Easy

517 of 791

Understanding +4 | -1

The value of

\int\limits_{0}^x d[t],

View answer and explanation

Correct option: D

[x]

Option D is correct because the integral of the differential of the greatest integer function [t] from 0 to x is simply [x].

518 Easy

518 of 791

Analyzing +4 | -1

Let

I = \int \nolimits_0^1 \,\,\frac{{\sin \,\,x}}{{\sqrt x }}\,\,dx\,\,\mathrm{and}\,\,J = \int \nolimits_0^1 \,\,\frac{{\cos \,\,x}}{{\sqrt x }}\,\,dx

. Then, which one of the following is true?

View answer and explanation

Correct option: C

I < \frac{2}{3}\,\,\mathrm{and}\,\,J < 2

Option C is correct because, for x in (0,1), sin(x) < x and cos(x) < 1, allowing us to bound the integrals I and J, resulting in I < 2/3 and J < 2.

519 Easy

519 of 791

Application +4 | -1

\int_a^b\{f(x)-3 x\} d x=a^2-b^2

, then the value of

f\left(\frac{\pi}{6}\right)

, is

View answer and explanation

Correct option: A

\frac{\pi}{6}

The correct answer is A (

\frac{\pi}{6}

) because after evaluating the integral and differentiating, we find that

f(x) = x

, thus

f(\frac{\pi}{6}) = \frac{\pi}{6}

520 Easy

520 of 791

Application +4 | -1

\lim \limits_\limits {n \rightarrow \infty} \frac{1^p+2^p+3^p+\ldots .+n^p}{n^{p+1}}

View answer and explanation

Correct option: A

\frac{1}{p+1}

Option A (

\frac{1}{p+1}

) is correct because the limit of the sum is evaluated as a definite integral of

x^p

from 0 to 1.

521 Easy

521 of 791

Application +4 | -1

\int\limits_{0}^{\pi / 2}|\sin x-\cos x| d x=

View answer and explanation

Correct option: B

2(\sqrt{2}-1)

Option B (

2(\sqrt{{2}}-1)

) is correct because it is the result of evaluating the definite integral of the absolute difference between sine and cosine functions over the given interval.

522 Easy

522 of 791

Analyzing +4 | -1

Let

I=\int_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x

. Then

View answer and explanation

Correct option: C

I > \sqrt 3 - \frac{\pi }{6}

The correct answer is C (

I > \sqrt 3 - \frac{\pi }{6}

) because the integral I is bounded below by the given expression, which is derived by using the minimum value of

\frac{{\sin x}}{x}

in the interval.

523 Easy

523 of 791

Application +4 | -1

\lim\limits_{n \to \infty } \,\,\frac{{{1^{99}} + \,\,{2^{99}}\,\, + \,\, \ldots + \,\,{n^{99}}}}{{{n^{100}}}} =

View answer and explanation

Correct option: B

\frac{{1}}{{100}}

Option B (

\frac{{1}}{{100}}

) is correct because the limit of the sum can be expressed as a definite integral, which evaluates to

\frac{1}{100}

524 Easy

524 of 791

Application +4 | -1

The integral

\int\limits_{{\pi / 4}}^{3 \pi / 4} \frac{1}{1+\cos x} d x

, is equal to

View answer and explanation

Correct option: D

Option D (2) is correct because by using the property

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

and trigonometric identities, the integral simplifies to 2.

525 Easy

525 of 791

Application +4 | -1

Area traced by the region "

M

" where

M=\left\{(x, y) / y \leq 1+\sqrt{x}, x \geq 0, y \leq 4^{\frac{x}{9}}\right\}

View answer and explanation

Correct option: A

27 \log _4\left(\frac{4}{\mathrm{e}}\right)

Option A (

27 \log _4\left(\frac{{4}}{{\mathrm{e}}}\right)

) is correct because it is the area calculated by integrating the difference between the two functions from 0 to 9.

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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

526 Easy

526 of 791

Application +4 | -1

Let

f(x)

be a function satisfying

f^{\prime}(x)=f(x)

with

f(0)=1

and

g(x)

be a function which satisfies

f(x)+g(x)=x^2

. Then

\int\limits_{0}^1 f(x) g(x) d x=

View answer and explanation

Correct option: D

e-\frac{e^2}{2}-\frac{3}{2}

Option D (

e-\frac{e^2}{2}-\frac{3}{2}

) is correct because it is the result of integrating the product of f(x) and g(x) from 0 to 1, where f(x) = e^x and g(x) = x² - e^x.

527 Easy

527 of 791

Application +4 | -1

\int\limits_{0}^1 \tan ^{-1}\left(\frac{1}{x^2-x+1}\right) d x

View answer and explanation

Correct option: D

\frac{\pi}{2}-\ln 2

Option D (

\frac{{\pi}}{{2}}-\ln 2

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

528 Easy

528 of 791

Application +4 | -1

\int \nolimits_{\left[ x \right]}^{\left[ x \right] + 1} \,\,f\left( t \right)\,\,dt = \left[ x \right]

,〗 then the value of

\int \nolimits_{ - 2}^4 \,\,f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

\mathrm{3}

Option D (3) is correct because the integral from [x] to [x]+1 of f(t) dt equals [x], allowing us to sum the integrals from -2 to 4 to get 3.

529 Easy

529 of 791

Application +4 | -1

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

View answer and explanation

Correct option: A

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

Option A (-1/(2(e^(2x)+1))+c) is correct because it is the result of integrating the given function using substitution.

530 Easy

530 of 791

Application +4 | -1

Let

f

be an odd function then

\int\limits_{{-1}}^1(|x|+f(x) \cos x) \mathrm{d} x

is equal to

View answer and explanation

Correct option: B

Option B is correct because the integral simplifies to integrating the absolute value function, as the other term is an odd function integrated over a symmetric interval.

531 Easy

531 of 791

Application +4 | -1

\frac{1}{\sqrt{a}} \int\limits_{{1}}^{a}\left(\frac{3}{2} \sqrt{x}+1-\frac{1}{\sqrt{x}}\right) d x < 4

, then 'a' may take values :

View answer and explanation

Correct option: A

Option A (0) is correct because the inequality holds true for a value of 'a' approaching 0.

532 Easy

532 of 791

Analyzing +4 | -1

The integral

\int \limits_0^{\pi /2} \,\,f\,\,\left( {\sin \,\,2x} \right)\,\,\sin \,\,x\,\,dx

is equal to

View answer and explanation

Correct option: B

\int \limits_0^{\pi /2} \,\,f\left( {\sin \,\,2x} \right)\,\,\cos \,\,x\,\,dx = \sqrt 2 \,\,\int \limits_0^{\pi /4} \,\,f\left( {\cos \,\,\,2x} \right)\cos \,\,x\,\,dx

Option B is correct because after applying the property of definite integrals and substitution, the given integral is equal to

\sqrt 2 \,\,\int \limits_0^{\pi /4} \,\,f\left( {\cos \,\,\,2x} \right)\cos \,\,x\,\,dx

533 Easy

533 of 791

Application +4 | -1

The value of

\int\limits_{0}^{[x]}(-x-[x]) d x

, is

View answer and explanation

Correct option: C

(1 / 2)[x]

Option C (

(1 / 2)[x]

) is correct because the integral simplifies using the periodic property of the function

x-[x]

534 Easy

534 of 791

Understanding +4 | -1

Let

f

be a continuous function

[a, b]

such that

f(x) > 0

for all

x \in[a, b]

. If

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

then

View answer and explanation

Correct option: B

F

is differentiable and increasing on

[a, b]

Option B (

F

is differentiable and increasing on

[a, b]

) is correct because

F'(x) = f(x) > 0

[a, b]

, which means

F

is increasing and differentiable.

535 Easy

535 of 791

Analyzing +4 | -1

The value of

I=\int\limits_{0}^{\pi / 2} \frac{(\sin x+\cos x)^2}{\sqrt{1+\sin 2 x}} d x

, is

View answer and explanation

Correct option: C

2

The correct answer is C (

2

) because the integral simplifies to

\int_0^{\pi/2} (\sin x + \cos x) dx

, which evaluates to 2.

536 Easy

536 of 791

Application +4 | -1

Area of the region bounded by

x=0, y=0, x=2, y \leq e^x

and

y \geq \ln x

, is

View answer and explanation

Correct option: A

6-4 \ln 2

Option A is correct because it represents the area calculated by integrating ln(x) from 1 to 2 and subtracting it from the area of the rectangle.

537 Easy

537 of 791

Application +4 | -1

\int \nolimits_0^1 \,\,\frac{{x\,\,dx}}{{\left[ {x + \sqrt {1 - {x^2}} } \right]\,\,\sqrt {1 - {x^2}} }}

is equal to

View answer and explanation

Correct option: C

\mathrm{\pi }/4

Option C (

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

) is correct because after substituting

x = \sin \theta

and using properties of definite integrals, the integral simplifies to this value.

538 Easy

538 of 791

Application +4 | -1

\int \nolimits_0^1 \,\,{x^{3/2}}\,\,\sqrt {1 - x\,\,} dx

is equal to

View answer and explanation

Correct option: D

\frac{\pi }{16}

Option D (

\frac{\pi}{16}

) is correct because the integral is solved using a trigonometric substitution and Walli's formula.

539 Easy

539 of 791

Analyzing +4 | -1

I_1=\int\limits_{0}^1\left(1-x^{50}\right)^{100} d x, I_2=\int\limits_{0}^1\left(1-x^{50}\right)^{101}

and

I_2=\alpha I_1

, then

\alpha

equals to

View answer and explanation

Correct option: C

\frac{5050}{5051}

The correct answer is C (

\frac{5050}{5051}

) because by using integration by parts,

I_2

can be expressed in terms of

I_1

, leading to the given ratio.

540 Easy

540 of 791

Analyzing +4 | -1

f(n)=\frac{1}{n}\{(n+1)(n+2)(n+3) \ldots(n+n)\}^{1 / n}

, then

\lim \limits_{{n \rightarrow \infty}} f(n)

equals

View answer and explanation

Correct option: D

4 / e

Option A (less than

10^{-7}

) is correct because the integral's absolute value is less than

10^{-7}

due to the small magnitude of the integrand over the given interval.

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