Question

The value of $\underset{-1}{\overset{1}{\int }}\left(x-\left[x\right]\right)dx$ (where $\left[.\right]$ denotes greatest integer function) is

• A. 0
• B. 1
• C. 2
• D. None of these

Answer 1

Answer: (B)

$I=\underset{-1}{\overset{1}{\int }}\left(x-\left[x\right]\right)dx=\underset{-1}{\overset{1}{\int }}xdx-{\int }_{-1}^1\left[x\right]dx$
$={\left[\frac{x^2}{2}\right]}_{-1}^1-\left[\underset{-1}{\overset{0}{\int }}\left[x\right]dx+\underset{0}{\overset{1}{\int }}\left[x\right]dx\right]$
$=\frac{1}{2}\left[1-1\right]-\left[\underset{-1}{\overset{0}{\int }}\left(-1\right)dx+\underset{0}{\overset{1}{\int }}0.dx\right]$
$\left[\begin{array}{c}If-1\le x<0,\left[x\right]=-1\\ If0\le x<1,\left[x\right]=0\end{array}\right]$
$=0-{\left[-x\right]}_{-1}^0-0=0-\left[-0-\left(-1\right)\right]=1$