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 · 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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451 Easy

451 of 791

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.

452 Easy

452 of 791

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.

453 Easy

453 of 791

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.

454 Easy

454 of 791

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.

455 Easy

455 of 791

Application +4 | -1

\int\limits_{0}^\pi \sqrt{\frac{1+\cos 2 x}{2}} d x

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because the integral simplifies to the integral of |cos(x)| from 0 to pi, which equals 2.

456 Easy

456 of 791

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.

457 Easy

457 of 791

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.

458 Easy

458 of 791

Application +4 | -1

The area between the curve

y = x\;{e^x}

and

y = x\;{e^{ - x}}

and the line

x = 1

in square unit, is

View answer and explanation

Correct option: D

\frac{2}{e}

sq unit

Option D (

\frac{2}{e}

sq unit) is correct because the area between the curves is found by integrating the difference of the functions from 0 to 1.

459 Easy

459 of 791

Understanding +4 | -1

f(x)=\int\limits_{a}^x t^3 e^t d t

, then

\frac{d}{d x} f(x)=

View answer and explanation

Correct option: B

x^3 e^x

Option B (

x^3 e^x

) is correct because, by the Fundamental Theorem of Calculus, the derivative of the integral of a function is the function itself.

460 Easy

460 of 791

Application +4 | -1

The value of the integral

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

View answer and explanation

Correct option: C

-\pi / 2

Option C (

-\pi / 2

) is correct because it involves correctly evaluating the derivative and the definite integral, while accounting for the discontinuity of the arctangent function.

461 Easy

461 of 791

Application +4 | -1

\int_0^{1.5} x\left[x^2\right] d x=

(where [.] denotes Greatest Integer Function)

View answer and explanation

Correct option: C

0.75

Option C (0.75) is correct because the integral is evaluated by splitting the interval based on the greatest integer function values of

x^2

462 Easy

462 of 791

Application +4 | -1

\int\limits_{{-1}}^1|x| d x

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because the integral of |x| from -1 to 1 is the area under the curve, which is 1.

463 Easy

463 of 791

Application +4 | -1

\int\limits_{0}^\pi x f(\sin x) d x=A \int\limits_{0}^{\pi / 2} f(\sin x) d x

, then

A

View answer and explanation

Correct option: B

\pi

Option B (

\pi

) is correct because by using the property

\int_0^a f(x) dx = \int_0^a f(a-x) dx

, the integral transforms to

A \int_0^{\pi/2} f(\sin x) dx

, where A is found to be

\pi

464 Easy

464 of 791

Application +4 | -1

I=\int_0^{\pi / 2} \frac{\sin \theta}{\sin \theta+\cos \theta} d \theta

, then

I=

View answer and explanation

Correct option: C

\pi / 4

Option C (

\pi / 4

) is correct because the integral can be solved by using the property of definite integrals and adding the original integral to a modified version.

465 Easy

465 of 791

Application +4 | -1

\int \nolimits_0^1 \,\,\left( {\left\{ x \right\}\,\,\left[ {x + 1} \right]} \right)dx

(where

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

denotes fractional part of

x

and

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

denotes greatest integer of

x)

is equal to

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{1}{2}

) is correct because the integral simplifies to

\int_0^1 x dx

due to the properties of fractional and greatest integer functions.

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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

466 Easy

466 of 791

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}

is equal to

View answer and explanation

Correct option: C

The correct answer is 19 because the limit can be expressed as a definite integral, which can then be evaluated.

467 Easy

467 of 791

Application +4 | -1

The area bounded by

y = \left[ x \right]

and the two ordinates

x = 1\,and\,x = 1.7

View answer and explanation

Correct option: D

\frac{7}{{10}}

Option D,

\frac{7}{{10}}

, is the correct area because the greatest integer function [x] equals 1 between x = 1 and x = 1.7, so the area is simply the integral of 1 from 1 to 1.7.

468 Easy

468 of 791

Application +4 | -1

\int \nolimits_0^{\mathrm{\pi }/2} \,\,\frac{{{{\sin }^2}\,\,x}}{{{{\sin }^2}x\,\, + \,\,{{\cos }^2}x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

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

Option A (

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

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

469 Easy

469 of 791

Application +4 | -1

f(x)=\left\{\begin{array}{cc}e^{\cos x} \sin x & \text { for }|x| \leq 2 \\ 2 & \text { otherwise }\end{array}\right.

then

\int\limits_{{-2}}^3 f(x) d x=

View answer and explanation

Correct option: C

Option C is correct because the integral can be solved by splitting the integral and recognizing the symmetry of the first part.

470 Easy

470 of 791

Application +4 | -1

Area bounded by the curves

y=e^x, y=2 x-x^2

and the line

x=0, x=1

; is equal to

View answer and explanation

Correct option: D

\frac{3 e-5}{3}

sq. units

Option D is correct because it represents the area calculated by integrating the difference between e^x and (2x - x²) from 0 to 1.

471 Easy

471 of 791

Application +4 | -1

Let

5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0

. Then

18 \int\limits_{{1}}^{2} f(x) d x

is equal to

View answer and explanation

Correct option: C

10 \log _{e} 2-6

Option C (

10 \log _{e} 2-6

) is correct because it is the value of the definite integral after solving the functional equation.

472 Easy

472 of 791

Application +4 | -1

The value of

\int\limits_{0}^1 \sqrt{\frac{1-x}{1+x}} d x

, is

View answer and explanation

Correct option: B

\frac{\pi}{2}-1

Option B is correct. The integral is evaluated by rationalizing the denominator and then splitting the integral into two simpler integrals.

473 Easy

473 of 791

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: D

\mathrm{4}

Option D (4) is correct because the integral of the absolute value of the greatest integer function over the given interval evaluates to 4.

474 Easy

474 of 791

Analyzing +4 | -1

The value of

\lim \limits_{{x \rightarrow 1}} \frac{\int\limits_{0}^{(x-1)^2} t \cos t^2 d t}{(x-1) \sin (x-1)}

View answer and explanation

Correct option: A

is equal to 0

Option A (is equal to 0) is correct because applying L'Hopital's rule twice results in a limit of 0.

475 Easy

475 of 791

Application +4 | -1

I_n=\int_{-n}^n\left(\{x+1\}\left\{x^2+2\right\}+\left\{x^2+3\right\}\left\{x^3+4\right\}\right) d x

, (where, ff denotes the fractional part), then

I_1

is equal to

View answer and explanation

Correct option: B

-\frac{2}{3}

Option D (None of these) is correct because the integral evaluates to -2/3, which is not among the provided options. The integral is simplified using properties of fractional parts and symmetry.

476 Easy

476 of 791

Application +4 | -1

The value of

\int\limits_{0}^8|x-5| d x

View answer and explanation

Correct option: A

Option A (17) is correct because the integral of |x-5| from 0 to 8 is calculated by splitting the integral at x=5 and evaluating each part separately.

477 Easy

477 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) dx < 4

, then '

a

' may take values:

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the inequality simplifies to

a + \sqrt{a} - 6 < 0

, which leads to

a \in [0, 4)

, and none of the given options satisfy this condition while also being defined.

478 Easy

478 of 791

Application +4 | -1

\int\limits_{{-4}}^4|x+2| d x=

View answer and explanation

Correct option: C

Option C (20) is correct because it is the result of evaluating the definite integral of the absolute value function.

479 Easy

479 of 791

Application +4 | -1

f(x)=f(2-x)

, then

\int\limits_{{0.5}}^{1.5} x f(x) d x

equals

View answer and explanation

Correct option: B

\int\limits_{{0.5}}^{1.5} f(x) d x

Option B (

\int\limits_{{0.5}}^{1.5} f(x) d x

) is correct because by applying the property of definite integrals, the given integral simplifies to this expression.

480 Easy

480 of 791

Application +4 | -1

The value of the integral

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

View answer and explanation

Correct option: C

\mathrm{4}\,\mathrm{/}\,\mathrm{3}

Option C (

\frac{\mathrm{4}}{\mathrm{3}}

) is correct because the integral is solved by simplifying the expression under the square root and using substitution.

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