$\frac{d}{{dx}}\,\,\left( {\mathop \smallint \limits_{f\left( x \right)}^{g\left( x \right)} \,\,\phi \,\,\left( t \right)\,\,dt} \right)$ is equal to

$\phi \,\,\left( {\,g\left( x \right)} \right)\,\, - \,\,\phi \,\,\left( {\,f\left( x \right)} \right)$

$\frac{1}{2}\,\,{\left[ {\,\phi \left( {\,g\,\,\left( x \right)} \right)} \right]^2}\,\, - \,\,\frac{1}{2}\,\,{\left[ {\,\phi \left( {\,f\,\,\left( x \right)} \right)} \right]^2}$

$\phi '\,\,\left( {\,g\left( x \right)} \right)\,\,g'\,\,\left( x \right)\,\, - \,\,\phi '\,\,\left( {f\left( x \right)} \right)\,\,f'\,\,\left( x \right)$

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

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

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

If $f(x)=f(a-x)$, then $\int_\limits0^a x f(x) d x$ equals

$\frac{a}{2} \int_\limits0^a x f(x) d x$

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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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Showing 421-450 of 791 questions

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421 Easy

421 of 791

Application +4 | -1

f

and

g

are continuous functions on

[0, \pi]

satisfying

f(x)+f(\pi-x)=1=g(x)+g(\pi-x), \quad

then

\int_0^\pi\{f(x)+g(x)\} d x

is equal to

View answer and explanation

Correct option: A

\pi

Option A (

\pi

) is correct because it uses the given functional equations to simplify the integral.

422 Easy

422 of 791

Application +4 | -1

\frac{d}{{dx}}\,\,\left( {\int \limits_{f\left( x \right)}^{g\left( x \right)} \,\,\phi \,\,\left( t \right)\,\,dt} \right)

is equal to

View answer and explanation

Correct option: C

g'\left( x \right)\,\,\phi \left( {\,g\,\,\left( x \right)} \right)\,\, - \,\,f'\left( x \right)\,\,\phi \left( {\,f\,\,\left( x \right)} \right)

Option C (

g'(x)\,\,\phi (\,g\,\,( x ) )\, - \,f'(x)\,\,\phi (\,f\,\,( x ) )

) is correct because it directly applies the Leibniz rule for differentiating an integral with variable limits.

423 Easy

423 of 791

Application +4 | -1

Value of

x

satisfying

\int\limits_{{\log 2}}^x \frac{d x}{\sqrt{e^x-1}}=\frac{\pi}{6}

View answer and explanation

Correct option: C

2 \log 2

Option C (

2 \log 2

) is correct because solving the integral equation

\int_{\log 2}^x \frac{{dx}}{{\sqrt{{e^x-1}}}} = \frac{{\pi}}{{6}}

leads to

x = \log 4 = 2 \log 2

424 Easy

424 of 791

Application +4 | -1

If [.] stands for the greatest integer function, then

\int\limits_{1}^2[3 x] d x

is equal to

View answer and explanation

Correct option: B

Option B (4) is correct because the integral is evaluated by breaking it into intervals where the greatest integer function has a constant value.

425 Easy

425 of 791

Analyzing +4 | -1

I_1=\int\limits_{e}^{e^2} \frac{d x}{\log x}

and

I_2=\int\limits_{1}^2 \frac{e^x}{x} d x

, then

View answer and explanation

Correct option: A

I_1=I_2

Option A (

I_1=I_2

) is correct because by substituting

\log x = t

I_1

, we obtain the same integral as

I_2

426 Easy

426 of 791

Analyzing +4 | -1

{I_n} = \int \nolimits_0^{\pi /4} \,\,{\tan ^n}\,\,x\,\,dx

, then

\lim\limits_{n \to \infty } n\,\,\left[ {{I_n}\,\, + \,\,{I_{n + 2}}} \right]

is equal to

View answer and explanation

Correct option: B

\mathrm{1}

Option B (

\mathrm{1}

) is correct because the limit is evaluated by simplifying the integral and applying the limit definition.

427 Easy

427 of 791

Analyzing +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left\{\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{3 n}\right\}=

View answer and explanation

Correct option: B

\log _e 3

Option B (

\log _e 3

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

\log _e 3

428 Easy

428 of 791

Application +4 | -1

The value of

\int \limits_{1/e}^e \,\,\frac{{|\log \,\,x|}}{{{x^2}}}\,\,dx

, is

View answer and explanation

Correct option: C

2\,\,\left( {1\,\, - \,\,\frac{1}{e}} \right)

Option C,

2\,\,\left( {1\,\, - \,\,\frac{1}{e}} \right)

, is the correct answer because it is the result of evaluating the definite integral with the absolute value of the logarithm.

429 Easy

429 of 791

Application +4 | -1

The value of

\int\limits_{1}^5(|x-3|+|1-x|) d x

View answer and explanation

Correct option: D

Option D (12) is correct because the integral evaluates to 12 after splitting the integral based on the absolute value functions and integrating each part.

430 Easy

430 of 791

Application +4 | -1

The value of the integral

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

, where

\left[ \cdot \right]

and

\left\{ \cdot \right\}

denote respectively the greatest integer function and fractional part function, is equal to

View answer and explanation

Correct option: B

\frac{{5 - \sqrt 5 }}{2}

Option B (

\frac{{5 - \sqrt 5 }}{2}

) is correct because evaluating the integral of the greatest integer function and fractional part function over the given limits results in this value.

431 Easy

431 of 791

Analyzing +4 | -1

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

[.] denotes the greatest integer function, is equal to

View answer and explanation

Correct option: D

-\frac{\pi}{2}

Option D (

-\frac{\pi}{2}

) is correct because the integral evaluates to this value after using the property of definite integrals and the greatest integer function.

432 Easy

432 of 791

Analyzing +4 | -1

The value of

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

is equal to

View answer and explanation

Correct option: A

\frac{\pi^2}{4}-2

Option A (

\frac{\pi^2}{4}-2

) is correct because the integral evaluates to this value after applying integration by parts and properties of definite integrals.

433 Easy

433 of 791

Application +4 | -1

Let

I_n=\int\limits_{0}^{\pi / 2} x^n \cos x \mathrm{~d} x

then

I_8+56 I_6

is equal to

View answer and explanation

Correct option: B

\left(\frac{\pi}{2}\right)^8

Option B ((

\frac{π}{2}

)⁸) is correct because it uses integration by parts to find a recursive relationship for I_n and then simplifies the expression I_8 + 56I_6.

434 Easy

434 of 791

Application +4 | -1

Let

f: R \in R

be a continuous function given by

f(x+y)=f(x)+f(y)

for all

x, y \in R

.
If

\int_0^2 f(x) d x=\alpha

, then

\int_{-2}^2 f(x) d x

is equal to

View answer and explanation

Correct option: C

The correct answer is C (0) because the given functional equation implies that f(x) is of the form

\lambda x

, making it an odd function, and the integral of an odd function over a symmetric interval is zero.

435 Easy

435 of 791

Analyzing +4 | -1

The value of the integral

\int\limits_{0}^1 e^{x^2} d x

lies in the interval

View answer and explanation

Correct option: C

(1, \mathrm{e})

Option C ((1, e)) is correct because the integral is bounded by the minimum and maximum values of the function within the given interval.

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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

436 Easy

436 of 791

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.

437 Easy

437 of 791

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.

438 Easy

438 of 791

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.

439 Easy

439 of 791

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

440 Easy

440 of 791

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.

441 Easy

441 of 791

Application +4 | -1

\int\limits_{0}^8 f(x) d x=3

then

\sum\limits_{{k=1}}^8 \int_0^1 f(k-1+x) d x

View answer and explanation

Correct option: A

Option A (3) is correct because the sum of the integrals can be combined into a single integral over the interval [0, 8], which is given as 3.

442 Easy

442 of 791

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})

443 Easy

443 of 791

Application +4 | -1

\int\limits_{{\pi / 4}}^{\pi / 2} \cot x d(\cos x)=

View answer and explanation

Correct option: B

\frac{1-\sqrt{2}}{\sqrt{2}}

Option B (

\frac{{1-\sqrt{{2}}}}{{\sqrt{{2}}}}

) is correct because the integral is evaluated by substituting

\cot x = \frac{{\cos x}}{{\sin x}}

and simplifying.

444 Easy

444 of 791

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.

445 Easy

445 of 791

Application +4 | -1

The value of

\int\limits_{0}^{\sqrt{2}}\left[x^2\right] d x

, where [.] is the greatest integer function

View answer and explanation

Correct option: C

\sqrt{2}-1

Option C (

\sqrt{2}-1

) is correct because the integral can be evaluated by splitting it into intervals where the greatest integer function has a constant value.

446 Easy

446 of 791

Application +4 | -1

\int\limits_{0}^{\infty} \frac{d x}{\left(x+\sqrt{x^2+1}\right)^3}=

View answer and explanation

Correct option: A

\frac{3}{8}

Option A (3/8) is correct because the integral evaluates to 3/8 after substituting x = tan(theta) and simplifying.

447 Easy

447 of 791

Application +4 | -1

f(x)=f(a-x)

, then

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

equals

View answer and explanation

Correct option: D

None of these

None of these is the correct option. The correct answer is

\frac{{a}}{{2}} \int_0^a f(x) dx

because using the substitution

x = a - u

and the given property

f(x) = f(a-x)

leads to this result.

448 Easy

448 of 791

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.

449 Easy

449 of 791

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.

450 Easy

450 of 791

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.

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