$\mathop \smallint \nolimits_1^2 \,\,{e^x}\,\,\left( {\frac{1}{x}\,\, - \,\,\frac{1}{{{x^2}}}} \right)\,\,dx$ is equal to

$e\,\, - \,\,\frac{{{e^2}}}{2}$

If ${I_1} = \mathop \smallint \limits_0^1 \,{2^{{x^2}}}\,dx,\,\;{I_2} = \mathop {\smallint \,}\limits_0^1 {2^{{x^3}}}\,dx,\;\,{I_3} = \mathop \smallint \limits_1^2 \,{2^{{x^2}}}dx\,\,{\rm{and}}\,\,{I_4} = \mathop \smallint \limits_1^2 \,{2^{{x^3}}}\,dx$ Then

${I_2} > {I_1}$ and ${I_3} > {I_4}$

If $F\,\,\left( x \right) = \mathop \smallint \limits_{{x^2}}^{{x^3}} \,\,\log \,\,t\,\,dt\,\,(x > 0)$, then $F'\left( x \right)$ equals

$\left( {4x\,\, - \,\,9{x^2}} \right)\,\,\log \,\,x$

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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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721 Easy

721 of 791

Application +4 | -1

The value of the integral

\int\limits_{0}^{\pi / 2} \frac{\mathrm{d} x}{1+\frac{1}{6} \sin ^2 x} is

View answer and explanation

Correct option: A

\frac{\pi}{2} \sqrt{\frac{6}{7}}

Option A is correct because the integral is solved by converting the integrand using trigonometric identities and then applying a standard integral formula.

722 Easy

722 of 791

Understanding +4 | -1

\int\limits_{{-1}}^1 \sin ^{11} x d x

is equal to

View answer and explanation

Correct option: D

Option D is correct because sin¹¹(x) is an odd function, and the integral of an odd function over a symmetric interval is zero.

723 Easy

723 of 791

Analyzing +4 | -1

Let

f:[-2,3] \rightarrow[0, \infty)

be a continuous function such that

f(1-x)=f(x)

for all

x \in[-2,3]

.
If

R_1

is the numerical value of the area of the region bounded by

y=f(x), x=-2, x=3

and the axis of

x

and

R_2=\int_{-2}^3 x f(x) d x

, then:

View answer and explanation

Correct option: D

\mathrm{R}_1=2 \mathrm{R}_2

Option D (

\mathrm{R}_1=2 \mathrm{R}_2

) is correct because by using the property of definite integrals and the given condition

f(1-x)=f(x)

, we can show that

2R_2 = R_1

724 Easy

724 of 791

Analyzing +4 | -1

16 \int_{1}^{2} \frac{d x}{x^{3}\left(x^{2}+2\right)^{2}}

is equal to

View answer and explanation

Correct option: C

\frac{11}{6}-\ln 4

Option C (

\frac{{11}}{{6}}

-\ln 4) is correct because after performing the integration and substituting the limits, the result matches this option.

725 Easy

725 of 791

Application +4 | -1

\int\limits_{0}^{\frac{\pi}{4}}[\sin x+\cos x] d(x-[x])

, [.] greatest integer function, equals

View answer and explanation

Correct option: A

\frac{\pi}{4}

Option A (

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

) is correct because the integral simplifies to

\int_0^{\frac{{\pi}}{{4}}} d\{x\}

, which evaluates to

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

726 Easy

726 of 791

Application +4 | -1

Area under the curve

y=\sin 2 x+\cos 2 x

between

x=0

and

x=\frac{\pi}{4}

, is

View answer and explanation

Correct option: B

1 sq. unit

Option B (1 sq. unit) is correct because the area is calculated by integrating

\sin 2x + \cos 2x

from

x=0

x=\frac{{\pi}}{{4}}

727 Easy

727 of 791

Application +4 | -1

The value of

\int\limits_{a}^b \frac{x}{|x|} d x, a < b < 0

View answer and explanation

Correct option: B

|b|-|a|

Option B (

|b|-|a|

) is correct because when a < b < 0, x/|x| = -1, and the integral becomes -(b-a) which simplifies to |b| - |a|.

728 Easy

728 of 791

Application +4 | -1

The area of the region bounded by the curve

y=\frac{1}{1+(\tan x)^{1.2}}

and the

x

-axis between the ordinates

x=\frac{\pi}{6}

and

x=\frac{\pi}{3}

View answer and explanation

Correct option: B

\frac{\pi}{12}

Option B (

\frac{{\pi}}{{12}}

) is correct because it correctly evaluates the definite integral using the property of definite integrals.

729 Easy

729 of 791

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: B

\frac{{{e^2}}}{2}\,\, - \,\,e

Option B is correct because after applying integration by parts, the integral simplifies to the given expression.

730 Easy

730 of 791

Application +4 | -1

A line tangent to the graph of the function

y=f(x)

at the point

x=a

forms an angle

\pi / 3

with the axis of abscissas and an angle

\pi / 4

at the point

x=b

then

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

is (

f^{\prime \prime}

is assumed to be a continuous function)

View answer and explanation

Correct option: B

1-\sqrt{3}

Option B (

1-\sqrt{3}

) is correct because it is the difference between the tangent of the angles at points b and a, which corresponds to the difference in the first derivative of the function at those points.

731 Easy

731 of 791

Application +4 | -1

\int \nolimits_a^b \,\,\sqrt {\left( {x - a} \right)\,\,\left( {b - x} \right)} \;dx,\;\,(b > a)

is equal to

View answer and explanation

Correct option: A

\frac{{\pi {{\left( {b\,\, - \,\,a} \right)}^2}}}{8}

The correct answer is A (

\frac{{\pi {{\left( {b\, - \,a} \right)}^2}}}{8}

) because after applying the trigonometric substitution and evaluating the definite integral, the result matches option A.

732 Easy

732 of 791

Application +4 | -1

\int\limits_{0}^3[x] d x

is equal to

View answer and explanation

Correct option: C

Option C (3) is correct because the integral of the greatest integer function [x] from 0 to 3 is equal to 3.

733 Easy

733 of 791

Application +4 | -1

Let

f

be a continuous function on

\mathbf{R}

satisfying

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

for all

x, y \in \mathbf{R}

with

f(1)

=2

and

g

be a function satisfying

f(x)+g(x)=

e^x

then the value of the integral

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

View answer and explanation

Correct option: C

2 / 3

Option C (2/3) is correct because after determining

f(x) = 2x

and

g(x) = e^x - 2x

, the integral

\int_0^1 f(x)g(x) dx

evaluates to 2/3.

734 Easy

734 of 791

Application +4 | -1

The value of

\lim\limits_{n \to \infty } \frac{1}{n}\sum\limits_{r = 0}^{2n - 1} {\frac{{{n^2}}}{{{n^2} + 4{r^2}}}}

View answer and explanation

Correct option: B

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

Option B is correct because the limit of the Riemann sum can be expressed as a definite integral, which evaluates to

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

735 Easy

735 of 791

Application +4 | -1

\int \nolimits_{ - 1/7}^{20/7} \,\,\sin \,\,\left( {x - \left[ x \right]} \right)\,\,dx

(where

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

denotes greatest integer) is

View answer and explanation

Correct option: C

3\,\,(1 - \,\,\cos \,\,1)

Option C (

3\,\,(1 - \,\,\cos \,\,1)

) is correct because the integral is evaluated using the property of greatest integer function and then integrating.

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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

736 Easy

736 of 791

Application +4 | -1

\int_e^x t f(t) d t=\sin x-x \cos x-\frac{x^2}{2}

for all

x \in R-\{0\}

, then the value of

f\left(\frac{\pi}{6}\right)

will be equal to

View answer and explanation

Correct option: C

-\frac{1}{2}

Option C (-1/2) is correct because differentiating the given integral equation using Leibniz's rule allows us to solve for f(x), and then evaluating f(π/6) yields -1/2.

737 Easy

737 of 791

Analyzing +4 | -1

{I_n} = \int \limits_0^{\pi /4} {\tan ^n}\,\,x\,\,dx

, then

\lim\limits_{n \to \infty } \,\,n\,\,\left( {{I_{n + 1}}\,\, + \,\,{I_{n - 1}}} \right)

equals

View answer and explanation

Correct option: A

Option A (1) is correct because the limit simplifies to 1 by using the recurrence relation In+1 + In-1 = 1/n.

738 Easy

738 of 791

Application +4 | -1

If the tangent to the graph function

y = f\left( x \right)

makes angles

\frac{\pi }{4}

and

\frac{\pi }{3}

with the

x

-axis is at the point

x = 2

and

x = 4

respectively, the value of

\int \nolimits_2^4 \,f'\left( x \right)\,\,f''\left( x \right)\,\,\;dx

View answer and explanation

Correct option: D

\mathrm{1}

Option D (

\mathrm{1}

) is correct because it is the result of evaluating the definite integral using the given tangent values.

739 Easy

739 of 791

Analyzing +4 | -1

\int\limits_{0}^\pi \log (1+\cos x) d x=

View answer and explanation

Correct option: D

-2 \pi \log 2

The correct answer is C (-

\pi \log 2

) because evaluating the definite integral using properties of logarithms and definite integrals leads to this result.

740 Easy

740 of 791

Analyzing +4 | -1

{I_1} = \int \limits_0^1 \,{2^{{x^2}}}\,dx,\,\;{I_2} = \mathop {\int \,}\limits_0^1 {2^{{x^3}}}\,dx,\;\,{I_3} = \int \limits_1^2 \,{2^{{x^2}}}dx\,\,\mathrm{and}\,\,{I_4} = \int \limits_1^2 \,{2^{{x^3}}}\,dx

Then

View answer and explanation

Correct option: A

{I_1} > {I_2}

and

{I_4} > {I_3}

Option C (

{I_1} > {I_2}

and

{I_3} > {I_4}

) is correct because for 0 < x < 1, x² > x³, and for 1 < x < 2, x³ > x².

741 Easy

741 of 791

Analyzing +4 | -1

For

I(x)=\int \frac{\sec ^2 x-2022}{\sin ^{2022} x} d x

, if

I\left(\frac{\pi}{4}\right)=2^{1011}

, then

View answer and explanation

Correct option: A

3^{1010} I\left(\frac{\pi}{3}\right)-I\left(\frac{\pi}{6}\right)=0

Option A (

3^{1010} I(\frac{\pi}{3})-I(\frac{\pi}{6})=0

) is correct because after integrating and substituting the given values, the equation holds true.

742 Easy

742 of 791

Application +4 | -1

The value of the integral

\int \limits_0^3 \,\,\frac{{dx}}{{\sqrt {x + 1} \, + \sqrt {5x + 1} }}

View answer and explanation

Correct option: D

None of these

The correct answer is D (None of these) because the integral evaluates to a different expression than the provided options.

743 Easy

743 of 791

Application +4 | -1

Let

f(x)=\int\limits_{1}^x \sqrt{2-t^2} d t

. Then the real roots of the equation

x^2-f^{\prime}(x)=0

are

View answer and explanation

Correct option: A

\pm 1

Option A (

\pm 1

) is correct because these values satisfy the equation

x^2 = \sqrt{2-x^2}

derived from the given conditions.

744 Easy

744 of 791

Application +4 | -1

Area bounded by the curve

x y-3 x-2 y-10=0, x

-axis and the lines

x=3, x=4

View answer and explanation

Correct option: C

16 \log 2+3

Option D (None of these) is incorrect. Option C (

16 \log 2+3

) is correct because the area is calculated by integrating the given function from x=3 to x=4, which results in

3+16 \log 2

745 Easy

745 of 791

Application +4 | -1

\int\limits_{{-\pi}}^\pi(\cos p x-\sin q x)^2 \mathrm{~d} x

, where

p

and

q

are integers is equal to

View answer and explanation

Correct option: D

2 \pi

Option D (8) is correct because by solving the given equation for [x/8], we find the interval for x and then calculate b-a.

746 Easy

746 of 791

Application +4 | -1

The value of

\int_{-\pi / 4}^{\pi / 4}\left(x|x|+\sin ^3 x+x \tan ^2 x+1\right) d x \text { is }

View answer and explanation

Correct option: D

\pi / 2

Option D (

\pi / 2

) is correct because the integral of the odd functions over the symmetric interval is zero, and the integral of 1 from

-\pi/4

\pi/4

\pi/2

747 Easy

747 of 791

Application +4 | -1

F\,\,\left( x \right) = \int \limits_{{x^2}}^{{x^3}} \,\,\log \,\,t\,\,dt\,\,(x > 0)

, then

F'\left( x \right)

equals

View answer and explanation

Correct option: A

\left( {9\,{x^2}\,\, - \,\,4\,x} \right)\,\,\log \,\,x

Option A is correct because the derivative of the integral is found using Leibniz's rule, resulting in (9x² - 4x)log x.

748 Easy

748 of 791

Analyzing +4 | -1

The value of the integral

\int \limits_0^{2\pi } \,\,\frac{{\sin \,\,2\theta }}{{a - b\,\,\cos \,\,\theta }}\,\,d\theta

when

a\,\,\, > \,\,\,b\,\,\, > \,\,\,0

, is

View answer and explanation

Correct option: D

\mathrm{1}

Option A is correct because the integral evaluates to 0 due to the symmetry of the integrand over the interval [0, 2π].

749 Easy

749 of 791

Application +4 | -1

The value of

\int\limits_{0}^{2 \pi} \cos ^{99} x d x

View answer and explanation

Correct option: D

Option D (0) is correct because the integral of an odd power of cosine over a symmetric interval around the y-axis is zero.

750 Easy

750 of 791

Analyzing +4 | -1

The value of

\int\limits_{{-2}}^2 \frac{3 x^2}{1+e^x} d x

, is

View answer and explanation

Correct option: A

Option A (8) is correct because the integral simplifies to

\int_0^2 3x^2 dx

using the property

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

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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