Question

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

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

Answer 1

Answer: (B)

${\int }_{-1}^1(x-[x])dx={\int }_{-1}^{1}xdx-{\int }_{-1}^1[x]dx$
$=[\frac{x^2}{2}{]}_{-1}^1-[{\int }_{-1}^0[x]dx+{\int }_0^1[x]dx]$
$=\frac{1}{2}[1-1]-[{\int }_{-1}^0(-1)dx+{\int }_0^{1}0.dx]$
[If -1 $\ge$ x < 0, [x] = -1
If -0 $\ge$ 0 < 1, [x] = -1 ]
$=0-[-x{]}_{-1}^0-0=0-[-0-(1)]=1$