Question

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

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

Answer 1

Answer: (B)

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