Answer

Verified

340.2k+ views

**Hint:**Whenever we have this type of question, one thing we need to remember is that the greatest integer function $\left[ x \right]$ breaks at integers which are usually discontinuous. So now we need to break the given limits as $ - 1 \to 0$ and $0 \to 1$ then do the integration to arrive at the required answer.

**Complete step by step answer:**

Here in this type of problems, finding the integration of the greatest integer function. Greatest integer function is also known as step function or floor function. One thing we need to remember is that the greatest integer function $\left[ x \right]$ breaks at integers which are usually discontinuous. Hence we can divide the given limit $ - 1 \to 1$ into two separate limit as $ - 1 \to 0$ and $0 \to 1$. therefore we can write the given function which is \[\int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx\] as below.

\[\int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = \int\limits_{ - 1}^0 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx + \int\limits_0^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx\]

Whenever we have limit $ - 1 \to 0$ the value inside the box or braces becomes $ - 1$ and in case of limit $0 \to 1$ we get the value inside the box or braces as $0$, which can be written as below.

\[ \Rightarrow \int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = \int\limits_{ - 1}^0 {\left[ {x + \left[ {x + \left[ { - 1} \right]} \right]} \right]} dx + \int\limits_0^1 {\left[ {x + \left[ {x + \left[ 0 \right]} \right]} \right]} dx\]

\[ \Rightarrow \int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = \int\limits_{ - 1}^0 {\left[ { - 1 + \left[ { - 1 + \left[ { - 1} \right]} \right]} \right]} dx + \int\limits_0^1 {\left[ {0 + \left[ {0 + \left[ 0 \right]} \right]} \right]} dx\]

\[ \Rightarrow \int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = \int\limits_{ - 1}^0 { - 3} dx + \int\limits_0^1 0 dx\]

Now integrate the function, we get

\[ \Rightarrow \int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = \left[ { - 3x} \right]_{ - 1}^0 + 0\]

Now apply the limit, and simplify the expression. We get

\[ \Rightarrow \int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = - 3(0 - ( - 1)) + 0\]

\[ \Rightarrow \int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx = - 3(1) + 0 = - 3\]

**Hence the integration of greatest integer function \[\int\limits_{ - 1}^1 {\left[ {x + \left[ {x + \left[ x \right]} \right]} \right]} dx\] is $ - 3$. Therefore, the option C is the correct answer.**

**Note:**

Whenever we have this type of problems, first we need to know the concept of greatest integer function, integration and simplifying the limits. And when integrating the function you should be clear with the integration concepts then only you can get the correct answer, also when simplifying the limits be careful.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

Using the following information to help you answer class 12 chemistry CBSE

Why should electric field lines never cross each other class 12 physics CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Distinguish between the reserved forests and protected class 10 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give simple chemical tests to distinguish between the class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

Which of the following books is not written by Harshavardhana class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

In which states of India are mango showers common What class 9 social science CBSE

What Made Mr Keesing Allow Anne to Talk in Class class 10 english CBSE