The value of the integral $\mathop \smallint \limits_1^3 \,\,\sqrt {3\,\, + \,\,{x^3}} \,\,dx$ lies in the interval

$\begin{array}{*{20}{l}} {\left( {{\rm{1,}}\,\,{\rm{3}}} \right)} \end{array}$

The value of $\mathop \smallint \nolimits_0^1 \,\,\frac{{{{\tan }^{ - 1}}\,\,x}}{{1 + \,\,{x^2}}}\;\,\,dx$ is

$\frac{{{{\rm{\pi }}^2}}}{{32}}$

$I = \left| {\,\mathop \smallint \nolimits_2^5 \,\,\frac{{\sin \,\,x\,\,dx}}{{(1 + {x^2}}}\,} \right|,\,\,{\rm{then}}$

$I$ lies in the interval $\left( {\frac{1}{4},\,\,\frac{1}{5}} \right)$

$I$ liesin the interval $\left( {\frac{1}{5},\,\,\frac{1}{6}} \right)$

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

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

For any natural number $n$, the value of the integral $\mathop \smallint \limits_0^{\sqrt n } \,\,\left[ {{x^2}} \right]\,\,dx$, is

$n\,\sqrt n \,\, - \,\,\mathop \sum \limits_{r = 1}^n \,\sqrt r $

$n\,\sqrt n \,\, + \,\,\mathop \sum \limits_{r = 1}^n \,\sqrt r $

$\mathop \sum \limits_{r = 1}^n \,\,\sqrt r \,\, - \,\,n\,\sqrt n $

If $f\left( x \right) = A\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right) + B,\;\,\,f'\,\,\left( {\frac{1}{2}} \right) = \sqrt 2 $ and $\mathop \smallint \limits_0^1 \,f\left( x \right)dx = \frac{{2\,A}}{\pi }\,$, then the constants $A$ and $B$ are respectively

$\frac{4}{\pi }$ and ${\rm{0}}$

$\frac{\pi }{2}$ and $\frac{\pi }{2}$

$\frac{2}{\pi }$ and $\frac{3}{\pi }$

${\rm{0}}$ and $ - \frac{4}{\pi }$

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JEE • MATHEMATICS • DEFINITE INTEGRATION

Definite integration practice · Mixed difficulty

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901 Medium

901 of 995

Analyzing +4 | -1

The value of the integral

\int \limits_1^3 \,\,\sqrt {3\,\, + \,\,{x^3}} \,\,dx

lies in the interval

View answer and explanation

Correct option: C

\left( {4,\,\,2\,\,\sqrt {300} } \right)

The correct answer is C (4, 2

\sqrt{30}

) because the integral is bounded by the minimum and maximum values of the integrand over the interval.

902 Easy

902 of 995

Application +4 | -1

The value of

\int \nolimits_0^1 \,\,\frac{{{{\tan }^{ - 1}}\,\,x}}{{1 + \,\,{x^2}}}\;\,\,dx

View answer and explanation

Correct option: B

\frac{{{\mathrm{\pi }^2}}}{{32}}

Option B (

\frac{{{\mathrm{\pi }^2}}}{{32}}

) is correct because it is the result of a u-substitution and evaluating the resulting integral.

903 Easy

903 of 995

Analyzing +4 | -1

The function

F\left( x \right) = \int \limits_0^x \log \,\,\left( {\frac{{1 - x}}{{1 + x}}} \right)dx

, is

View answer and explanation

Correct option: A

An even function

Option A (An even function) is correct because the integral of an odd function from 0 to x results in an even function.

904 Medium

904 of 995

Application +4 | -1

\int\limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^{2}}} d x

is equal to

View answer and explanation

Correct option: D

2 \pi

Option D (

2 \pi

) is correct because it is the result of evaluating the definite integral.

905 Medium

905 of 995

Analyzing +4 | -1

Let

f

be integrable over

\left[ {0,\,\,a} \right]

for any real

a

. If we define

{I_1} = \int \nolimits_0^{\pi /2} \,\,\cos \,\,\mathrm{\theta }\,\,f(\sin \,\,\mathrm{\theta }\,\, + \,\,{\cos ^2}\,\,\mathrm{\theta })\,\,\;d\mathrm{\theta }

and

{I_2} = \int \nolimits_0^{\pi /2} \sin \,\,2\mathrm{\theta }\,\,f(\sin \,\,\mathrm{\theta }\,\, + \,\,{\cos ^2}\,\,\mathrm{\theta })\;\,\,d\mathrm{\theta }

, then

View answer and explanation

Correct option: A

{I_1} = {I_2}

Option A is correct because by substituting sin(

\theta

) + cos²(

\theta

) = t, the integral I1 - I2 becomes zero, implying I1 = I2.

906 Easy

906 of 995

Application +4 | -1

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

is equal to

View answer and explanation

Correct option: A

\frac{2}{\sqrt{5}} \tan ^{-1} \frac{1}{\sqrt{5}}

Option A (

\frac{2}{\sqrt{5}} \tan ^{-1} \frac{1}{\sqrt{5}}

) is correct because using the Weierstrass substitution simplifies the integral to a standard arctangent form.

907 Easy

907 of 995

Application +4 | -1

The value of the integral

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

is equal to

View answer and explanation

Correct option: B

2 \pi

Option B (

2 \pi

) is correct because the expression inside the integral simplifies to

\frac{\pi}{2}

, and the integral of a constant is straightforward.

908 Easy

908 of 995

Application +4 | -1

The value of

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

View answer and explanation

Correct option: C

\mathrm{\pi }/4

Option C (

\mathrm{\pi }/4

) is correct because it is obtained by using the property of definite integrals and adding two forms of the integral to simplify and solve.

909 Easy

909 of 995

Analyzing +4 | -1

I = \left| {\,\int \nolimits_2^5 \,\,\frac{{\sin \,\,x\,\,dx}}{{(1 + {x^2}}}\,} \right|,\,\,\mathrm{then}

View answer and explanation

Correct option: D

I \le \frac{3}{{10}}

Option D (

I \le \frac{3}{{10}}

) is correct because the integral is bounded above by

\frac{3}{10}

by using properties of inequalities and integrals.

910 Medium

910 of 995

Application +4 | -1

If the area above the x-axis bounded by the curves

y = {2^{kx}}\mathrm{\;and\;}x = 0\;\,\mathrm{and\;}\,2\;is\,\frac{3}{{\log 2\;}}\;\mathrm{then\;the\;value\;of\;}k\mathrm{\;is}

View answer and explanation

Correct option: B

Option B, 1, is the correct value of k because when k = 1, the area under the curve y = 2^(kx) from x = 0 to x = 2 equals 3/log(2).

911 Medium

911 of 995

Analyzing +4 | -1

The value of the integral

I=\int\limits_{0}^a \frac{x^4}{\left(a^2+x^2\right)^4} d x

, is

View answer and explanation

Correct option: A

\frac{1}{16 a^3}\left(\frac{\pi}{4}-\frac{1}{3}\right)

Option A (

\frac{1}{16 a^3}(\frac{\pi}{4}-\frac{1}{3})

) is correct because it is obtained by using trigonometric substitution and simplifying the integral.

912 Medium

912 of 995

Application +4 | -1

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

View answer and explanation

Correct option: C

\frac{\pi}{4}

Option C (

\frac{\pi}{4}

) is correct because using the property

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

simplifies the integral to a readily solvable form.

913 Medium

913 of 995

Application +4 | -1

If f(x)=f(a+x) and

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

is equal to

View answer and explanation

Correct option: A

n\,k

Option A (n k) is correct because the integral from 0 to na of f(x) dx is n times the integral from 0 to a of f(x) dx, given that f(x) = f(a+x).

914 Hard

914 of 995

Application +4 | -1

Let

{A_1}

be the area of the parabola

{y^2} = 4ax

lying between vertex and latusrectum and

{A_2}

be the area between latusrectum and double ordinate

x = 2a

. Then,

{A_1}/{A_2} =

View answer and explanation

Correct option: B

\left( {2\sqrt 2 + 1} \right)/7

Option B is correct because the ratio of the areas A1 and A2 is calculated using definite integrals and simplification, resulting in (2

\sqrt{2}

+ 1)/7.

915 Easy

915 of 995

Application +4 | -1

The value of

\int \nolimits_0^a \,\,\sqrt {\frac{{a - x}}{x}} \,\,dx

View answer and explanation

Correct option: C

\frac{{\pi a}}{2}

Option C (

\frac{{\pi a}}{2}

) is correct. The integral is solved using trigonometric substitution, leading to the value

\frac{\pi a}{2}

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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

916 Easy

916 of 995

Application +4 | -1

f(x)=[|x|]

, then

\int\limits_{0}^{50} f(x) d x

is equal to (where [.] denotes the greatest integer function)

View answer and explanation

Correct option: A

1225

Option A (1225) is correct because the integral is the sum of the integers from 0 to 49, which equals 1225.

917 Medium

917 of 995

Application +4 | -1

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {x,\,\;\;for\;\,\,x\,\, < 1}\\ {x - 1,\;\,\,for\;\,\,x\,\, \ge 1} \end{array}} \right.

, then

\int \limits_0^2 \,\,{x^2}\,\,f\left( x \right)dx

is equal to

View answer and explanation

Correct option: C

\frac{5}{3}

Option C (

\frac{5}{3}

) is correct because it is the result of correctly evaluating the definite integral by splitting it into two integrals based on the piecewise definition of f(x).

918 Easy

918 of 995

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.

919 Easy

919 of 995

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.

920 Hard

920 of 995

Application +4 | -1

The value of the integnal

\int\limits_{0}^3\left(x^2+1\right) \, d[x]

, is where

[\cdot]

is the greatest integer function

View answer and explanation

Correct option: C

Option C (17) is correct because the integral is evaluated using integration by parts and the properties of the greatest integer function.

921 Medium

921 of 995

Analyzing +4 | -1

The value of

\int\limits_{0}^{\pi / 2}(2 \log \sin x-\log \sin 2 x) d x

, is

View answer and explanation

Correct option: B

-\frac{\pi}{2} \log 2

Option B (

\frac{-\pi}{2} \log 2

) is correct because it is found by using properties of logarithms and definite integrals to simplify and evaluate the integral.

922 Medium

922 of 995

Understanding +4 | -1

f(x)

is a continuous periodic function with period

T

, then the integral

I=\int\limits_{a}^{a+T} f(x) d x

View answer and explanation

Correct option: C

Independent of a

The correct answer is Independent of a because the integral of a periodic function over one period is constant regardless of the starting point.

923 Easy

923 of 995

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.

924 Medium

924 of 995

Application +4 | -1

\lim \limits_{{n \rightarrow \infty}}\left[\sin \frac{n}{n^2+1^2}+\sin \frac{n}{n^2+1^2}+\ldots+\sin \frac{n}{n^2+1^2}\right]=

View answer and explanation

Correct option: A

\frac{\pi}{4}

Option A (

\frac{\pi}{4}

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

\frac{\pi}{4}

925 Easy

925 of 995

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.

926 Medium

926 of 995

Application +4 | -1

For any natural number

n

, the value of the integral

\int \limits_0^{\sqrt n } \,\,\left[ {{x^2}} \right]\,\,dx

, is

View answer and explanation

Correct option: B

n\,\sqrt n \,\, - \,\,\sum \limits_{r = 1}^n \,\sqrt r

Option D (

n\sqrt n - \sum \limits_{r = 1}^n \,\sqrt r

) is correct because the integral is evaluated by splitting it into a sum of integrals and simplifying.

927 Easy

927 of 995

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.

928 Medium

928 of 995

Application +4 | -1

f\left( x \right) = A\,\,\sin \,\,\left( {\frac{{\pi x}}{2}} \right) + B,\;\,\,f'\,\,\left( {\frac{1}{2}} \right) = \sqrt 2

and

\int \limits_0^1 \,f\left( x \right)dx = \frac{{2\,A}}{\pi }\,

, then the constants

A

and

B

are respectively

View answer and explanation

Correct option: D

\frac{4}{\pi }

and

\mathrm{0}

Option D (

\frac{4}{\pi }

and 0) is correct. By using the given derivative and integral conditions, we can solve for the constants A and B.

929 Easy

929 of 995

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.

930 Easy

930 of 995

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.

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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