# Let $\left[ x \right]$denotes the greatest integer less than or equal to x, then the value of the integral $\int\limits_{ - 1}^1 {\left( {\left| x \right| - 2\left[ x \right]} \right)dx} $ is equal to

$

(a){\text{ 3}} \\

(b){\text{ 2}} \\

(c){\text{ - 2}} \\

(d){\text{ - 3}} \\

$

Last updated date: 16th Mar 2023

•

Total views: 303.9k

•

Views today: 7.82k

Answer

Verified

303.9k+ views

Hint – In this question we have to evaluate the integral and the functions involved inside are modulus function and greatest integer function. Firstly break the limits that are instead of directly having from -1 to 1 have them from -1 to 1 the form 0 to 1. The greatest integer function is defined only for intervals with stride of 1 unit. Then evaluate the integral.

Complete step-by-step answer:

Given integral is

$I = \int\limits_{ - 1}^1 {\left( {\left| x \right| - 2\left[ x \right]} \right)dx} $, where $\left[ x \right]$ denotes greatest integer function.

Now as we know in the interval $\left( { - \infty ,0} \right)$ the value of the greatest integer function is -1.

And in the interval $\left( {0,\infty } \right)$ the value of the greatest integer function is zero (0).

So break the integration limits and put the value of greatest integer function.

$I = \int\limits_{ - 1}^0 {\left( {\left| x \right| - 2\left( { - 1} \right)} \right)dx} + \int\limits_0^1 {\left( {\left| x \right| - 2\left( 0 \right)} \right)dx} $

$ \Rightarrow I = \int\limits_{ - 1}^0 {\left( {\left| x \right| + 2} \right)dx} + \int\limits_0^1 {\left( {\left| x \right|} \right)dx} $

Now as we know that the value of $\left| x \right|$ is -x on L.H.S and +x on R.H.S

I.e. $\left\{ \begin{gathered}

\left| x \right| = - x,{\text{ }}x \in \left[ { - 1,0} \right) \\

\left| x \right| = x,{\text{ }}x \in \left( {0,1} \right] \\

\end{gathered} \right.$

So, substitute this value in above equation we have,

$ \Rightarrow I = \int\limits_{ - 1}^0 {\left( { - x + 2} \right)dx} + \int\limits_0^1 {x{\text{ }}dx} $

Now apply the integration we have,

$ \Rightarrow I = \left[ {\dfrac{{ - {x^2}}}{2} + 2x} \right]_{ - 1}^0 + \left[ {\dfrac{{{x^2}}}{2}} \right]_0^1$

Now apply the integration limit we have,

$ \Rightarrow I = \left[ {0 + 0 - \left( {\dfrac{{ - {{\left( { - 1} \right)}^2}}}{2} + 2\left( { - 1} \right)} \right)} \right] + \left[ {\dfrac{{{1^2}}}{2} - 0} \right]$

Now simplify the above equation we have,

$ \Rightarrow I = \dfrac{1}{2} + 2 + \dfrac{1}{2} = 1 + 2 = 3$

So this is the required value of the integral.

Hence option (a) is correct.

Note – Whenever we face type of questions the key concept is to have the understanding of modulus function and greatest integer function. The basic definition of modulus function is $\left\{ \begin{gathered}

\left| x \right| = - x,{\text{ }}x < 0 \\

\left| x \right| = x,{\text{ }}x \geqslant 0 \\

\end{gathered} \right.$. Its plot is a pair of straight lines passing through origin and changing the slopes at point (0, 0). A greatest integer function is denoted by less than or equal to x for $\left[ x \right]$, it basically rounds down a real number to the nearest integer. These concepts along with some basic integration properties will help you get on the right track to reach the answer.

Complete step-by-step answer:

Given integral is

$I = \int\limits_{ - 1}^1 {\left( {\left| x \right| - 2\left[ x \right]} \right)dx} $, where $\left[ x \right]$ denotes greatest integer function.

Now as we know in the interval $\left( { - \infty ,0} \right)$ the value of the greatest integer function is -1.

And in the interval $\left( {0,\infty } \right)$ the value of the greatest integer function is zero (0).

So break the integration limits and put the value of greatest integer function.

$I = \int\limits_{ - 1}^0 {\left( {\left| x \right| - 2\left( { - 1} \right)} \right)dx} + \int\limits_0^1 {\left( {\left| x \right| - 2\left( 0 \right)} \right)dx} $

$ \Rightarrow I = \int\limits_{ - 1}^0 {\left( {\left| x \right| + 2} \right)dx} + \int\limits_0^1 {\left( {\left| x \right|} \right)dx} $

Now as we know that the value of $\left| x \right|$ is -x on L.H.S and +x on R.H.S

I.e. $\left\{ \begin{gathered}

\left| x \right| = - x,{\text{ }}x \in \left[ { - 1,0} \right) \\

\left| x \right| = x,{\text{ }}x \in \left( {0,1} \right] \\

\end{gathered} \right.$

So, substitute this value in above equation we have,

$ \Rightarrow I = \int\limits_{ - 1}^0 {\left( { - x + 2} \right)dx} + \int\limits_0^1 {x{\text{ }}dx} $

Now apply the integration we have,

$ \Rightarrow I = \left[ {\dfrac{{ - {x^2}}}{2} + 2x} \right]_{ - 1}^0 + \left[ {\dfrac{{{x^2}}}{2}} \right]_0^1$

Now apply the integration limit we have,

$ \Rightarrow I = \left[ {0 + 0 - \left( {\dfrac{{ - {{\left( { - 1} \right)}^2}}}{2} + 2\left( { - 1} \right)} \right)} \right] + \left[ {\dfrac{{{1^2}}}{2} - 0} \right]$

Now simplify the above equation we have,

$ \Rightarrow I = \dfrac{1}{2} + 2 + \dfrac{1}{2} = 1 + 2 = 3$

So this is the required value of the integral.

Hence option (a) is correct.

Note – Whenever we face type of questions the key concept is to have the understanding of modulus function and greatest integer function. The basic definition of modulus function is $\left\{ \begin{gathered}

\left| x \right| = - x,{\text{ }}x < 0 \\

\left| x \right| = x,{\text{ }}x \geqslant 0 \\

\end{gathered} \right.$. Its plot is a pair of straight lines passing through origin and changing the slopes at point (0, 0). A greatest integer function is denoted by less than or equal to x for $\left[ x \right]$, it basically rounds down a real number to the nearest integer. These concepts along with some basic integration properties will help you get on the right track to reach the answer.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?