If ${I_1} = \mathop \smallint \nolimits_0^{\pi /2} \,\,x\,\,\sin \,\,x\,dx$ and ${I_2} = \mathop \smallint \nolimits_0^{\pi /2} \,x\,\,\cos \,\,x\,dx$, then which one of the following is true?

${I_2} - {I_1} = \frac{\pi }{2}$

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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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331 Easy

331 of 995

Application +4 | -1

Let

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

. Then,

\alpha=

View answer and explanation

Correct option: A

\ln 2

The correct answer is A (

\ln 2

) because after evaluating the integral and solving for

\alpha

, the value of

\alpha

is found to be

\ln 2

332 Medium

332 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: D

\frac{\pi}{4}

Option D (

\frac{\pi}{4}

) is correct because by using the property of definite integrals and adding the original integral to a transformed one, we simplify and solve for the integral's value.

333 Easy

333 of 995

Application +4 | -1

\int\limits_{{-a}}^a(|x|+|x-2|) d x=22

and

a>2

, then the value of

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

View answer and explanation

Correct option: A

Option A (3) is correct because the value of 'a' is found by solving the given integral equation, and then the second integral is evaluated using the periodic property.

334 Easy

334 of 995

Application +4 | -1

The value of

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

View answer and explanation

Correct option: D

\frac{\pi}{4}

Option D (

\frac{\pi}{4}

) is correct because the integral is evaluated by using a trigonometric substitution, which simplifies the integral to a known form.

335 Easy

335 of 995

Analyzing +4 | -1

{I_1} = \int \nolimits_0^{3\pi } \,\,f\,\,({\cos ^2}\,\,x)\,\,dx

and

{I_2} = \int \nolimits_0^\pi \,\,f({\cos ^2}\,\,x)\,\,dx,\mathrm{then}

,〗 then

View answer and explanation

Correct option: C

{I_1} = 3\,{I_2}

Option C (

{I_1} = 3\,{I_2}

) is correct because the integral

{I_1}

can be split into three equal integrals, each equal to

{I_2}

336 Easy

336 of 995

Analyzing +4 | -1

The value of the definite integral

\int\limits_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} {\frac{{dx}}{{(1 + \mathrm{ }{e^{x\,\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}}

is equal to

View answer and explanation

Correct option: B

\frac{\pi }{{2\sqrt 2 }}

Option B (

\frac{\pi }{{2\sqrt 2 }}

) is correct because it is the result of evaluating the definite integral after applying properties of definite integrals and trigonometric substitutions.

337 Medium

337 of 995

Application +4 | -1

The value of

I = \int \nolimits_0^1 \,x\,\,\left| {x\,\, - \,\,\frac{1}{2}} \right|\,dx

View answer and explanation

Correct option: C

\frac{1}{8}

Option C (

\frac{1}{8}

) is correct because it is the result of evaluating the definite integral with the absolute value function.

338 Medium

338 of 995

Application +4 | -1

\int \frac{x^2-1}{\left(x^2+1\right) \sqrt{x^4+1}}dx

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because after algebraic manipulation and substitution, the integral can be transformed into a standard form involving the inverse secant function.

339 Hard

339 of 995

Analyzing +4 | -1

Let

f\;:R \to R

be a continuous function such that

f\left( x \right)

is not identically equal to zero. If

\int \limits_2^5 \,\,f\left( x \right)\,\,dx = \int \limits_{ - 2}^5 \,\,f\left( x \right)\,\,dx

, then

f\left( x \right)

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given condition only implies that the integral from -2 to 2 is zero, which doesn't necessarily mean the function is odd or even.

340 Medium

340 of 995

Application +4 | -1

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

is a quadratic polynomial in x such that

6\,\,\int \limits_0^1 \,\,f\left( x \right)\,\,dx\,\, - \,\,\left\{ {f\left( 0 \right)\,\, + \,\,4f\,\,\left( {\frac{1}{2}} \right)} \right\} = k\,f\,\,\left( 1 \right)

, then

k =

View answer and explanation

Correct option: C

\mathrm{1}

Option C (

\mathrm{1}

) is correct because when

f(x)

is a quadratic polynomial, the given equation holds true only when

k = 1

341 Medium

341 of 995

Application +4 | -1

The value of

\int\limits_{0}^{\pi / 2} \frac{1+2 \cos x}{(2+\cos x)^2} \mathrm{~d} x

View answer and explanation

Correct option: C

1 / 2

Option C (

1 / 2

) is correct because the definite integral evaluates to 1/2 after applying trigonometric substitution and integration techniques.

342 Easy

342 of 995

Application +4 | -1

\int\limits_{0}^{\pi / 2} \frac{\cos x-\sin x}{1+\sin x \cos x} d x=

View answer and explanation

Correct option: C

Option C (0) is correct because the integral evaluates to zero due to the property of definite integrals where substituting

\frac{\pi}{2} - x

results in the negative of the original integrand.

343 Hard

343 of 995

Analyzing +4 | -1

\alpha=\int\limits_{0}^1 e^{9 x+3 \tan ^{-1} x}\left(\frac{12+9 x^2}{1+x^2}\right) d x

, where

\tan ^{-1} x

takes only principal values, then the value of

\left(\log _e|1+\alpha|-\frac{3 \pi}{4}\right)

, is

View answer and explanation

Correct option: A

Option A (9) is correct because it is found by substituting and simplifying the given integral and then taking the logarithm.

344 Easy

344 of 995

Application +4 | -1

The value of

\int\limits_0^{\frac{\pi }{2}} {\sqrt {\sin 2\theta } \sin \theta } \,d\theta

View answer and explanation

Correct option: D

\frac{\pi }{4}

Option D (

\frac{\pi }{4}

) is correct because the integral can be simplified using trigonometric identities and substitution to arrive at the final value.

345 Easy

345 of 995

Application +4 | -1

The value of integral,

: \int\limits_{3}^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x

View answer and explanation

Correct option: B

\frac{3}{2}

Option B (

\frac{3}{2}

) is correct because using the property

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

simplifies the integral.

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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

346 Easy

346 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

Option C (2) is correct because the integral of the absolute value function |1-x| over the interval [-1, 1] evaluates to 2.

347 Medium

347 of 995

Application +4 | -1

The integral

\int\limits_ {\frac{\pi}{12}}^{\frac{\pi}{4}} \frac{8 \cos 2 x}{(\tan x+\cot x)^3} d x

equals :

View answer and explanation

Correct option: A

\frac{15}{128}

Option A (

\frac{15}{128}

) is correct because it is the value obtained after simplifying the integral using trigonometric identities and evaluating the definite integral.

348 Easy

348 of 995

Application +4 | -1

\int \nolimits_0^{\pi /6} \frac{{\sin \,\,x}}{{{{\cos }^3}\,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{1}{6}

Option B (

\frac{1}{6}

) is correct because by substituting cos(x) = t, the integral simplifies and evaluates to 1/6.

349 Easy

349 of 995

Understanding +4 | -1

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

has the same value as the integral

View answer and explanation

Correct option: A

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

Option A (

\int_0^1 f(x) dx

) is correct because substituting 1-x = t transforms the integral into the same integral with variable x.

350 Easy

350 of 995

Application +4 | -1

Let

f\,\,\left( x \right) = x\,\, - \,\,\left[ x \right]

, for every real number

x

, where

\left[ x \right]

is the integral part of

x

. Then,

\int \nolimits_{ - 1}^1 \,\,f\,\,\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: A

1

Option A (1) is correct because the integral of the fractional part of x over the given interval is equal to 1.

351 Easy

351 of 995

Understanding +4 | -1

The value of

\int\limits_{0}^\pi \cos ^{11} x \,d x

, is

View answer and explanation

Correct option: C

Option C is correct because the integral of cos¹¹(x) from 0 to pi is zero due to the symmetry of the cosine function around pi/2.

352 Medium

352 of 995

Application +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: A

\frac{2^n}{m+1}-\frac{n}{m+1} I(m+1, n-1)

Option A is correct because it is derived using integration by parts, taking

u=(1+t)^n

and

dv=t^m dt

353 Medium

353 of 995

Analyzing +4 | -1

\int \nolimits_{ - 1/2}^{1/2} \,\,(\cos \,\,x)\left[ {\log \,\,\left( {\frac{{1 - x}}{{1 + x}}} \right)} \right]\,\,\,dx

is equal to

View answer and explanation

Correct option: A

\mathrm{0}

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

354 Easy

354 of 995

Analyzing +4 | -1

\left[ {\,\sum \limits_{n = 1}^{10} \,\,\int \limits_{ - 2n - 1}^{ - 2n} \,\,{{\sin }^{27}}\,\,x\,dx} \right]\,\, + \,\,\left[ {\sum \limits_{n = 1}^{10} \,\,\int \limits_{2n}^{2n + 1} \,\,{{\sin }^{27}}\,\,x\,dx} \right]

equals

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because the sum of the integrals can be shown to cancel out by using the substitution x = -t and the properties of the sine function.

355 Easy

355 of 995

Application +4 | -1

f\left( x \right)

be continuous function such that the area bounded by the curve

y = f\left( x \right)

, the

x

-axis and the lines

x = a\mathrm{\;and}\;x = 0\;\mathrm{is}\frac{{{a^2}}}{2} + \frac{a}{2}\sin a + \frac{\mathrm{\pi }}{2}\cos a.\,\mathrm{Value\;of\;}f\left( {\frac{\mathrm{\pi }}{2}} \right)\;\mathrm{is}

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (

\frac{1}{2}

) is correct because it is the result of differentiating the area function and substituting

\frac{\pi}{2}

for

a

356 Easy

356 of 995

Application +4 | -1

{I_1} = \int \nolimits_0^{\pi /2} \,\,x\,\,\sin \,\,x\,dx

and

{I_2} = \int \nolimits_0^{\pi /2} \,x\,\,\cos \,\,x\,dx

, then which one of the following is true?

View answer and explanation

Correct option: A

{I_1} + {I_2} = \frac{\pi }{2}

Option A (

{I_1} + {I_2} = \frac{\pi }{2}

) is correct because evaluating both integrals and summing them results in

\frac{\pi}{2}

357 Easy

357 of 995

Application +4 | -1

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

View answer and explanation

Correct option: B

Option B (0) is correct because the function

f(x) = x|x|

is an odd function, and the integral of an odd function over a symmetric interval [-a, a] is always zero.

358 Medium

358 of 995

Application +4 | -1

a\,\, < \,\,\int \nolimits_0^{2\pi } \,\,\frac{1}{{10\,\, + \,\,3\,\,\cos \,\,x}}dx\,\, < \,\,b

, then the ordered pair

\left( {a,\,\,b} \right)

View answer and explanation

Correct option: B

\left( {\frac{{2\pi }}{{13}},\,\,\frac{{2\pi }}{7}} \right)

Option B is correct because the integral is bounded by the minimum and maximum values of the integrand over the given interval, resulting in the interval (2π/13, 2π/7).

359 Medium

359 of 995

Application +4 | -1

\int \nolimits_1^3 \,\,\left( {x - 1} \right)\,\,\left( {x - 2} \right)\,\,\left( {x - 3} \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

\mathrm{0}

Option D (0) is correct because the integral evaluates to zero after expanding the polynomial and applying the limits of integration.

360 Easy

360 of 995

Application +4 | -1

The area between the curve

y = 4 + 3x - {x^2}

and

x

-axis is

View answer and explanation

Correct option: A

125/6 sq unit

Option A (125/6 sq unit) is correct because it represents the definite integral of the quadratic function from -1 to 4.

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