If ${I_1} = \mathop \smallint \limits_0^\infty \,\,\frac{1}{{1 + {x^4}}}\,\,dx$ and ${I_2} = \mathop \smallint \limits_0^\infty \,\,\frac{{{x^2}}}{{1\,\, + \,\,{x^4}}}\,\,dx$. Then $\frac{{{I_1}}}{{{I_2}}} = $

$\begin{array}{*{20}{l}} {{\rm{1}}\,{\rm{/}}\,{\rm{2}}} \end{array}$

If $f\left( x \right) = f\left( {a - x} \right),\;\,\,{\rm{then}}\,\,\mathop \smallint \nolimits_0^a \,\,f\left( x \right)\,\,dx$ is equal to

$\frac{a}{2}\,\,\mathop \smallint \nolimits_0^a \,\,f\left( x \right)\,\,dx$

$\mathop \smallint \nolimits_0^a \,\,f\left( x \right)\,\,dx$

$\frac{{{a^2}}}{2}\,\,\mathop \smallint \nolimits_0^a \,\,f\left( x \right)dx$

$ - \frac{a}{2}\,\,\mathop \smallint \nolimits_0^a \,\,f\left( x \right)\,\,dx$

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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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271 Medium

271 of 995

Application +4 | -1

\int f(x) d x=\psi(x)

, then

\int x^5 f\left(x^3\right) d x

is equal to

View answer and explanation

Correct option: C

\frac{1}{3} x^3 \psi\left(x^3\right)-\int x^2 \psi\left(x^3\right) d x+C

Option C (

\frac{1}{3} x^3 \psi(x^3)-\int x^2 \psi(x^3) d x+C

) is correct because integration by parts and substitution lead to this result.

272 Medium

272 of 995

Analyzing +4 | -1

{I_1} = \int \limits_0^\infty \,\,\frac{1}{{1 + {x^4}}}\,\,dx

and

{I_2} = \int \limits_0^\infty \,\,\frac{{{x^2}}}{{1\,\, + \,\,{x^4}}}\,\,dx

. Then

\frac{{{I_1}}}{{{I_2}}} =

View answer and explanation

Correct option: A

Option A (1) is correct because by substituting x = 1/t in I1, it transforms into I2, showing that I1 and I2 are equal.

273 Medium

273 of 995

Application +4 | -1

\int \limits_0^3 \,\,\left| {\,{x^3} + {x^2} + 3x\,} \right|\,\,\,dx

is equal to

View answer and explanation

Correct option: B

\frac{{171}}{4}

Option B (

\frac{{171}}{4}

) is correct because the integrand is positive over the interval [0, 3], and the integral evaluates to this value.

274 Easy

274 of 995

Application +4 | -1

The value of

\int\limits_{{-\pi}}^\pi \frac{\cos ^2 x}{1+a^x} d x, a>0

View answer and explanation

Correct option: C

\frac{\pi}{2}

Option C is correct because by using the property

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

and adding the two integrals, we can simplify the integral to

\int_{-\pi}^{\pi} \cos^2 x dx

which equals

\pi

, and hence I =

\frac{\pi}{2}

275 Medium

275 of 995

Application +4 | -1

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

View answer and explanation

Correct option: D

\pi / 4

Option D (

\pi / 4

) is correct because the integral simplifies to

\pi/4

after using the property of definite integrals and adding the original and transformed integrals.

276 Medium

276 of 995

Application +4 | -1

The area of the region bounded by the

\mathrm{x}

-axis and the curves defined by

y=\tan x,(-\pi / 3 \leq \mathrm{x} \leq \pi / 3)

View answer and explanation

Correct option: C

2 \log 2

Option C (

2 \log 2

) is correct because it represents the area of the region bounded by the x-axis and the curve y = tan(x) from -π/3 to π/3.

277 Easy

277 of 995

Application +4 | -1

2 \int\limits_{0}^1 \tan ^{-1} x \,d x=\int\limits_{0}^1 \cot ^{-1}\left(1-x+x^2\right) d x.

Then,

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

is equal to

View answer and explanation

Correct option: A

\log 2

Option A is correct. The equation is simplified using trigonometric identities and integration by parts, leading to the solution log 2.

278 Easy

278 of 995

Application +4 | -1

The value of the integral

\int\limits_{0}^{100 \pi} \sqrt{1-\cos 2 x}

dx is

View answer and explanation

Correct option: B

200 \sqrt{2}

Option B is correct because the integral is solved by using the identity for

\sqrt{1 - cos 2x}

and then integrating the absolute value of the sine function over the given interval.

279 Easy

279 of 995

Understanding +4 | -1

To find the numerical value of

\int \nolimits_{ - 2}^2 \,\,\left( {p{x^3} + qx + s} \right)\,\,dx,\,\,\mathrm{it}

is necessary to know the values of the constants

View answer and explanation

Correct option: C

s

Option C (s) is correct because the integral of px³ and qx from -2 to 2 is 0 (odd functions), so only the integral of s remains, which depends on the value of s.

280 Easy

280 of 995

Application +4 | -1

The area of the triangle formed by the positive

x

-axis and the normal and the tangent to the circle

x^2+y^2

=4

(1, \sqrt{3})

View answer and explanation

Correct option: A

2 \sqrt{3}

Option A (

2 \sqrt{3}

) is correct because the area of the triangle formed by the x-axis, tangent, and normal is

2\sqrt{3}

281 Medium

281 of 995

Application +4 | -1

\int\limits_{0}^{r / 2} \frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}} d x=

View answer and explanation

Correct option: C

\frac{\pi}{4}

Option C (

\frac{\pi}{4}

) is correct because by using the property of definite integrals and adding the original integral to the transformed integral, we can simplify the expression and solve for the integral.

282 Easy

282 of 995

Application +4 | -1

The value of

\int \nolimits_0^{12a} \,\,\frac{{f\left( x \right)}}{{f\left( x \right)\,\, + \,\,f\,\left( {12\,a - x} \right)}}dx\,\,\mathrm{is}

View answer and explanation

Correct option: D

6\,a

Option D (

6a

) is correct because by using the property of definite integrals, the integral simplifies to half the interval length, which is

6a

283 Easy

283 of 995

Application +4 | -1

\int\limits_{0}^1\left(1-x^2\right)^{3 / 2} d x=

View answer and explanation

Correct option: C

\frac{3 \pi}{16}

Option C (

\frac{3 \pi}{16}

) is correct because the integral evaluates to

\frac{3 \pi}{16}

after trigonometric substitution and simplification.

284 Easy

284 of 995

Analyzing +4 | -1

\lim\limits_{n \to \infty } \left[ {\frac{1}{n} + \frac{n}{{{{(n + 1)}^2}}} + \frac{n}{{{{(n + 2)}^2}}} + ... + \frac{n}{{{{(2n + 1)}^2}}}} \right]

equal to

View answer and explanation

Correct option: B

\frac{1}{2}\,

Option B (

\frac{1}{2}\,

) is correct because the limit of the summation can be converted into a definite integral, which evaluates to

\frac{1}{2}

285 Easy

285 of 995

Application +4 | -1

The value of

\int_{-\pi / 2}^{\pi / 2} \frac{1}{|x|+\mid \sin x]+4} d x

, where

[t]

denotes the greatest integer less than or equal to

t,

View answer and explanation

Correct option: D

\frac{3}{20}(4 \pi-3)

The correct answer is D (

\frac{3}{20}(4 \pi-3)

) because after evaluating the integral using the properties of the greatest integer function, the result is

\frac{3}{20}(4 \pi-3)

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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

286 Easy

286 of 995

Application +4 | -1

I = \int \limits_{ - 2}^2 \,\,\left| {1 - {x^4}} \right|\,\,dx

, then

\mathrm{\;I}

equals

View answer and explanation

Correct option: C

\mathrm{12}

Option C (12) is correct because the integral of the absolute value function |1 - x⁴| from -2 to 2 is calculated by splitting the integral at the points where 1 - x⁴ changes sign.

287 Medium

287 of 995

Application +4 | -1

f\left( x \right) = f\left( {a - x} \right),\;\,\,\mathrm{then}\,\,\int \nolimits_0^a \,\,f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{a}{2}\,\,\int \nolimits_0^a \,\,f\left( x \right)\,\,dx

Option C (

\frac{a}{2}\,\,\int \nolimits_0^a \,\,f(x)\,\,\;dx

) is correct because using the property

f(x) = f(a - x)

simplifies the integral to

\frac{a}{2}\,\,\int \nolimits_0^a \,\,f(x)\,\,\;dx

288 Easy

288 of 995

Analyzing +4 | -1

The value of integral

\int\limits_{ - 2}^2 {\frac{{\left| {{x^3} + x} \right|}}{{({e^{x\left| x \right|}} + 1)}}} dx

is equal to:

View answer and explanation

Correct option: D

Option D (6) is correct because the integral simplifies to the integral of an even function, which is then easily evaluated.

289 Easy

289 of 995

Application +4 | -1

\int {\frac{1}{x}\sqrt {\frac{{1 - x}}{{1 + x}}} dx = g\left( x \right) + c,\,\,g\left( 1 \right) = 0,}

then

g\left( {\frac{1}{2}} \right)

us equal to

View answer and explanation

Correct option: A

{\log _e}\left( {\frac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}} \right) + \frac{\pi }{3}

Option A (

{\log _e}\left( {\frac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}} \right) + \frac{\pi }{3}

) is correct because it is the result of the definite integral calculation with the given substitution and initial condition.

290 Medium

290 of 995

Application +4 | -1

a

and

b

are real numbers different from zero, then

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

View answer and explanation

Correct option: A

\frac{\pi}{2}

Option A (

\frac{\pi}{2}

) is correct because after simplification and using trigonometric substitutions, the integral evaluates to

\frac{\pi}{2}

291 Medium

291 of 995

Application +4 | -1

\int \nolimits_{ - 1/7}^{20/7} \,\,{e^{5\left\{ x \right\}}}\,\,dx

is equal to (where

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

denotes fractional part of

x

)

View answer and explanation

Correct option: B

\frac{3}{5}\,\,\left( {{e^5} - 1} \right)

Option B (

\frac{3}{5}\,\,\left( {{e^5} - 1} \right)

) is correct because the integral is evaluated by using the property of fractional part function and then integrating.

292 Easy

292 of 995

Application +4 | -1

The value of the integral

\int \limits_0^1 \,\,\frac{1}{{{{\left( {{x^2} + 1} \right)}^{3/2}}}}\,\,dx

, is

View answer and explanation

Correct option: B

\sqrt 2 /2

Option B (

\sqrt 2 /2

) is correct because the integral is solved using trigonometric substitution and simplification.

293 Easy

293 of 995

Application +4 | -1

{I_n} = \int \limits_0^\infty \,\,{e^{ - x}}\,{x^{n - 1}}\,\,dx

, then

\int \limits_0^\infty \,\,{e^{ - \lambda x}}\,\,{x^{n - 1}}\,\,dx

is equal to

View answer and explanation

Correct option: C

\frac{{{I_n}}}{{{\lambda ^n}}}

Option C (

\frac{{{I_n}}}{{{\lambda ^n}}}

) is correct because it is the result of a correct u-substitution and simplification.

294 Easy

294 of 995

Application +4 | -1

\int\limits_{0}^{\pi / 2} \max (\sin x, \cos x) d x

is equal to

View answer and explanation

Correct option: D

\sqrt{2}

Option D (

\sqrt{{2}}

) is correct because the integral

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

can be split into two integrals based on where

\sin x > \cos x

and vice versa, and the result is

\sqrt{{2}}

295 Easy

295 of 995

Application +4 | -1

\int\limits_{1}^2 \frac{d x}{\left(x^2-2 x+4\right)^{\frac{3}{2}}}=\frac{k}{k+5}

then

k

is equal to :

View answer and explanation

Correct option: A

Option A (1) is correct because it is the value of k that satisfies the given integral equation after evaluating the integral and solving for k.

296 Easy

296 of 995

Application +4 | -1

\int \nolimits_0^{\pi /2} \frac{{\cos \,\,x - \sin \,\,x}}{{1 + \,\,\cos \,\,x\,\,\sin \,\,x}}\,\,dx

is equal to

View answer and explanation

Correct option: A

\mathrm{0}

Option A (

\mathrm{0}

) is correct because using the property of definite integrals

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

leads to the integral being equal to its negative.

297 Easy

297 of 995

Application +4 | -1

The area of the loop between the curve

y = a\sin x

and

x -

axis is

View answer and explanation

Correct option: B

2a

Option B (

2a

) is correct because it represents the definite integral of

a\sin x

from 0 to

\pi

298 Medium

298 of 995

Application +4 | -1

For

n > 0

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

View answer and explanation

Correct option: C

\pi^2

Option C is correct because by using the property

\int_0^{2a} f(x) dx = \int_0^{a} f(x) dx + \int_0^{a} f(2a-x) dx

and simplifying, we can find the value of the integral to be

\pi^2

299 Easy

299 of 995

Application +4 | -1

\int \nolimits_a^b \,\,{x^3}\,\,\;dx = 0

and if

\int \nolimits_a^b \,\,{x^2}\;\,dx = \frac{2}{3}

, then the values of

a\;\,\,\mathrm{and}\,\,\;b

are respectively

View answer and explanation

Correct option: D

- 1,\,\,1

Option D (

- 1,\\,\\,1

) is correct because the integral of an odd function from -a to a is 0, and the integral of x² from -1 to 1 is 2/3.

300 Easy

300 of 995

Application +4 | -1

The equation of the tangent to the curve

y=\int\limits_{x}^{x^2} \log t \mathrm{~d} t \quad \text { at } x=2 \text { is }

View answer and explanation

Correct option: D

y+8 \log 2+2=(7 \log 2) x

The correct answer is D (

y+8 \log 2+2=(7 \log 2) x

) because it represents the equation of the tangent line to the given curve at x=2, found by evaluating the integral and its derivative.

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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