If $f\,\,\left( x \right) = f\,\,\left( {a + b - x} \right)$ for all $x\,\, \in \,\,\left[ {a,\,\,b} \right]$ and $\mathop {\smallint \,}\limits_a^b x{\rm{ }}\,f\,\left( x \right)\,\,dx = k\,\mathop \smallint \limits_a^b \,\,f\left( x \right)\,\,dx$, then the value of $k$ is

$\frac{{{a^2}\,\, + \,\,{b^2}}}{2}$

$\frac{{{a^2}\,\, - \,\,{b^2}}}{2}$

$\mathop \smallint \limits_0^\pi \,k{\left( {\pi \;x\,\, - \,\,{x^2}} \right)^{100}}\sin \,\,2x\,\,\;dx$ is equal to

$\frac{1}{2}\,\,\left( {{\pi ^{100}} - \,\,{\pi ^{97}}} \right)$

$\frac{1}{2}\,\,\left( {{\pi ^{100}} + \,\,{\pi ^{97}}} \right)$

The value of $\mathop \smallint \limits_a^{a + \left( {\pi /2} \right)} \,\,\left( {{{\sin }^4}\,\,x\,\, + \,\,{{\cos }^4}\,\,x} \right)\,\,dx$, is

$a\,\,{\left( {\frac{\pi }{2}} \right)^2}$

$ f(x)=\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+3\right)} d x $ if $ f(3)=\frac{1}{2}[\ln 5-\ln 6] $, then $ f(4) $ is:

$ \frac{1}{2}[\ln 19-\ln 17] $

If ${I_n} = \mathop \smallint \limits_0^{\pi /2} \,\,\frac{{{{\sin }^2}\,\,n\,\,{\rm{ }}x}}{{{{\sin }^2}\,\,x}}\,\,dx$, then which one of the following is not true?

${I_n} = 2\mathop \smallint \limits_0^{\pi /2} \,\,\frac{{\sin \,\,x\,\,\cos \,\,2n{\rm{ }}\,x}}{{\sin \,\,x}}\,\,dx$

${I_1},\,\,{I_2},\,\,{I_3},\,\, \ldots {I_n} \ldots .$ Is a A.P.

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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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571 Easy

571 of 995

Application +4 | -1

The integral

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

is equal to

View answer and explanation

Correct option: B

\tan ^{-1}(2)-\frac{\pi}{4}

Option B (tan^-1(2) - pi/4) is correct because after using the tangent half-angle substitution and integrating, the result is tan^-1(2) - pi/4.

572 Medium

572 of 995

Application +4 | -1

\int\limits_{0}^{\pi / 2} \frac{d x}{a^2 \cos ^2 x+{b}^2 \sin ^2 x}=\frac{\pi}{16}

. Then minimum value of

a \cos x+b \sin x

View answer and explanation

Correct option: A

-4

Option A (-4) is correct because the minimum value of the expression is found by relating the integral to ab and minimizing

a \cos x + b \sin x

using

a=b

573 Easy

573 of 995

Application +4 | -1

f\,\,\left( x \right) = f\,\,\left( {a + b - x} \right)

for all

x\,\, \in \,\,\left[ {a,\,\,b} \right]

and

\mathop {\int \,}\limits_a^b x\mathrm{ }\,f\,\left( x \right)\,\,dx = k\,\int \limits_a^b \,\,f\left( x \right)\,\,dx

, then the value of

k

View answer and explanation

Correct option: A

\frac{{a\,\, + \,\,b}}{2}\,

Option A is correct because by using the given property and substituting, the value of k is found to be the average of a and b.

574 Easy

574 of 995

Analyzing +4 | -1

The value of the integral

I = \int \limits_1^\infty \,\,\frac{{{x^2} - \,\,2}}{{{x^3}\,\sqrt {{x^2} - \,\,1} }}\,\,dx

, is

View answer and explanation

Correct option: A

0

Option A (0) is correct because the integral evaluates to zero due to a substitution that reveals symmetry.

575 Easy

575 of 995

Application +4 | -1

The integral

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

is equal to

View answer and explanation

Correct option: D

2\,\pi

Option D (

2\pi

) is correct because the integral simplifies to a constant value of

\pi/2

over the interval [-1, 3], resulting in

2\pi

576 Medium

576 of 995

Application +4 | -1

[x]

denotes the greatest integer less than or equal to

x

then the value of

\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}[[x]-\operatorname{Sin} x] d x=

View answer and explanation

Correct option: D

-\pi

Option D (

-\pi

) is correct because the integral simplifies to the integral of -1 over the given interval, which results in

-\pi

577 Easy

577 of 995

Analyzing +4 | -1

a = \lim\limits_{n \to \infty } \sum\limits_{k = 1}^n {\frac{{2n}}{{{n^2} + {k^2}}}}

and

f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}, x \in(0,1)

, then

View answer and explanation

Correct option: C

\sqrt{2} f\left(\frac{a}{2}\right)=f^{\prime}\left(\frac{a}{2}\right)

Option C (

\sqrt{2} f(\frac{a}{2})=f^{\prime}(\frac{a}{2})

) is correct because after calculating

a

and simplifying

f(x)

, we find that the given relation holds true.

578 Medium

578 of 995

Application +4 | -1

The value of

\int \nolimits_{ - 1}^1 \,\,\left[ {\,x\,} \right[1 + \sin \,\,\mathrm{\pi }x]\;dx

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

denotes the greatest integer)

View answer and explanation

Correct option: A

\mathrm{2}

Option A (2) is correct because the integral simplifies to 2 after considering the greatest integer function and the sine function within the given limits.

579 Medium

579 of 995

Analyzing +4 | -1

\int \nolimits_0^1 \,\,\log \,\,\left\{ {\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right)\,\,} \right\}\,\,dx

is equal to

View answer and explanation

Correct option: B

- \,\,\log \,\,2

Option B (

\ - \,\,\log \,2

) is correct because the integral can be evaluated using a known result for the integral of log sin(x).

580 Easy

580 of 995

Application +4 | -1

Let

f

be a continuous function satisfying

\int\limits_{{-\pi}}^{t^2}\left(f(x)+x^2\right) \mathrm{d} x=\pi+\frac{4}{3} t^3

for all

t

, then

f\left(\pi^2 / 4\right)

is equal to

View answer and explanation

Correct option: D

\pi-\frac{\pi^4}{16}

Option D (

\pi-\frac{\pi^4}{16}

) is correct because differentiating the given equation and substituting

t = \frac{\pi}{2}

yields the value of

f(\frac{\pi^2}{4})

581 Easy

581 of 995

Analyzing +4 | -1

\int \limits_0^\pi \,k{\left( {\pi \;x\,\, - \,\,{x^2}} \right)^{100}}\sin \,\,2x\,\,\;dx

is equal to

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because the integral evaluates to zero due to the symmetry properties after applying the substitution x = pi - x.

582 Medium

582 of 995

Application +4 | -1

The value of integral

\int \nolimits_{ - 1}^1 \,\,\frac{{\left| {\,x\,\, + \,\,2\,} \right|}}{{x\,\, + \,\,2}}\;\,\,dx

View answer and explanation

Correct option: B

\mathrm{2}

Option B (2) is correct because since x + 2 is positive over the interval [-1, 1], the absolute value simplifies, and the integral becomes

\int_{-1}^{1} 1 dx = 2

583 Easy

583 of 995

Application +4 | -1

The area below

y=e^x

and above

y=x

between

x=

0 and

x=2

View answer and explanation

Correct option: C

e^2-3

Option C (

e^2-3

) is correct because it is the result of integrating the difference between

e^x

and

x

from 0 to 2.

584 Easy

584 of 995

Analyzing +4 | -1

The integral

\int\limits_{{\pi / 6}}^{\pi / 3} \tan ^3 x \sin ^2 3 x\left(2 \sec ^2 x \sin ^2 3 x+3 \tan x \sin 6 x\right) d x

is equal to

View answer and explanation

Correct option: C

-\frac{1}{18}

Option C (

-\frac{1}{18}

) is correct because the integral simplifies to a direct evaluation of a function after recognizing the product rule in reverse.

585 Medium

585 of 995

Application +4 | -1

The value of

\int \limits_a^{a + \left( {\pi /2} \right)} \,\,\left( {{{\sin }^4}\,\,x\,\, + \,\,{{\cos }^4}\,\,x} \right)\,\,dx

, is

View answer and explanation

Correct option: A

\frac{{3\,\pi }}{8}

Option A is correct because the integral of a periodic function over one period is independent of the starting point, and the integral is evaluated to be 3π/8.

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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

586 Easy

586 of 995

Application +4 | -1

A(\mathrm{t})=\int\limits_{{-1}}^t e^{-|x|} d x

, then

\lim \limits_{{t \rightarrow \infty}} A(t)

is equal to

View answer and explanation

Correct option: A

2-e^{-1}

Option A (

2-e^{-1}

) is correct because it represents the limit of the integral as t approaches infinity, calculated by splitting the integral and evaluating the limits.

587 Easy

587 of 995

Application +4 | -1

\int\limits_{{-1}}^{3 / 2}|x \sin \pi x| \mathrm{d} x=k / \pi^2

, then the value of

k

View answer and explanation

Correct option: A

3 \pi+1

Option A (

3 \pi+1

) is correct because the integral of the absolute value function is split into intervals where the sign of the function inside the absolute value changes.

588 Medium

588 of 995

Application +4 | -1

The value of

\int\limits_{0}^2 x^{\left[x^2+1\right]} \mathrm{d} x

, where

[x]

is the greatest integer less than or equal to

x

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the integral evaluates to a more complex expression than the provided options.

589 Medium

589 of 995

Application +4 | -1

The value of

\int\limits_{0}^\pi x \log (\sin x) \mathrm{d} x

is (given that

\int\limits_{0}^{\pi / 2} \log \sin x \mathrm{~d} x=-\frac{\pi}{2} \log 2)

View answer and explanation

Correct option: A

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

Option B (6) is correct because by setting up equations based on the given information about pairs and triplets, we can solve for n.

590 Medium

590 of 995

Application +4 | -1

\lim\limits_{n \to \infty } \,\,\frac{{1\,\, + \,\,{2^4} + \,\,{3^4} + \,\,... + \,\,{n^4}}}{{{n^5}}}\,\, - \,\,\lim\limits_{n \to \infty } \,\,\frac{{1\,\, + \,\,{2^3}\,\, + \,\,{3^3}\,\, + \,\,... + \,\,{n^3}}}{{{n^5}}}

View answer and explanation

Correct option: D

\frac{1}{5}

Option D (

\frac{1}{5}

) is correct because it represents the difference between the integrals of x⁴ and 0, evaluated from 0 to 1.

591 Medium

591 of 995

Application +4 | -1

\int\limits_{0}^{10} e^{x-[x]} d x([\cdot]

denotes integral part

)

is equal to

View answer and explanation

Correct option: A

10(e-1)

Option A (10(e-1)) is correct because the integral simplifies to 10 times the integral from 0 to 1 of e^x, which evaluates to 10(e-1).

592 Medium

592 of 995

Application +4 | -1

\int \nolimits_0^\pi \,\,\frac{1}{{1 + \sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: B

The correct answer is 2 because it is the result of evaluating the definite integral using trigonometric substitution and limits.

593 Easy

593 of 995

Analyzing +4 | -1

The value of the integral

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

View answer and explanation

Correct option: B

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

594 Easy

594 of 995

Application +4 | -1

\int \nolimits_0^1 \,\,\left\{ x \right\}\,\,d\,\,\left\{ x \right\}

is equal to (where

\left\{ {\, \cdot \,} \right\}

denotes fractional part of

x

)

View answer and explanation

Correct option: C

\frac{1}{{2}}

Option C (

\frac{1}{{2}}

) is correct because the fractional part of x is x in the interval [0,1], and the integral of x from 0 to 1 is 1/2.

595 Medium

595 of 995

Application +4 | -1

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

f(3)=\frac{1}{2}[\ln 5-\ln 6]

, then

f(4)

is:

View answer and explanation

Correct option: A

\frac{1}{2}[\ln 17-\ln 19]

Option A (

\frac{{1}}{{2}}[\ln 17 - \ln 19]

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

596 Easy

596 of 995

Analyzing +4 | -1

{I_n} = \int \limits_0^{\pi /2} \,\,\frac{{{{\sin }^2}\,\,n\,\,\mathrm{ }x}}{{{{\sin }^2}\,\,x}}\,\,dx

, then which one of the following is not true?

View answer and explanation

Correct option: D

\sin \,\,\left( {{I_{15}}} \right) = 0

Option D (sin(I_15) = 0) is correct because it is not necessarily true, as I_n = n*pi/2 and sin(15*pi/2) is not 0.

597 Easy

597 of 995

Analyzing +4 | -1

I = \int \nolimits_0^{\pi /2} \,\,\mathrm{cos}\,\,(\sin \,\,x)\,\;dx

J = \int \nolimits_0^{\pi /2} \,\,\sin \,\,(\cos \,\,x)\;dx\;\,\,\mathrm{and\;}\,\,K = \int \nolimits_0^{\pi /2} \cos \,\,x\,dx.\,\mathrm{Then},

View answer and explanation

Correct option: D

I > K > J

Option D (

I > K > J

) is correct because, within the interval [0, π/2], cos(sin x) > cos x > sin(cos x), leading to the given inequality of integrals.

598 Easy

598 of 995

Application +4 | -1

The inlegral

\int\limits_{2}^4 \frac{\log x^2}{\log x^2+\log \left(36-12 x+x^2\right)} d x,

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because the integral is of the form

\int_a^b \frac{f(x)}{f(x) + f(a+b-x)} dx

, which simplifies to

\frac{b-a}{2}

, resulting in 1.

599 Medium

599 of 995

Application +4 | -1

The value of

\int\limits_{{-\pi / 2}}^{\pi / 2} \cos t \sin (2 t-\pi / 4) \mathrm{d} t

View answer and explanation

Correct option: D

-\sqrt{2} / 3

Option D (-

\sqrt{2}

/ 3) is correct because it is the result of evaluating the definite integral after applying trigonometric identities and simplifying.

600 Medium

600 of 995

Analyzing +4 | -1

The value of

\int \limits_0^{\pi /2} \,\,\frac{{x\,\, + \,\,\sin \,\,x}}{{1\,\, + \,\,\cos \,\,x}}\,\,dx

, is

View answer and explanation

Correct option: C

\pi /2

The correct answer is C (

\pi /2

) because the integral can be solved by splitting the integrand and using integration by parts.

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