Question

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

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

Answer 1

Answer: (B)

$\displaystyle \int _{- 1}^{1} \left(\right. x - \left[\right. x \left]\right. \left.\right) d x = \displaystyle \int _{- 1}^{1} x d x - \displaystyle \int _{- 1}^{1} \left[\right. x \left]\right. d x$
$= \left[\right. \frac{x^{2}}{2} \left]\right._{- 1}^{1} - \left[\right. \displaystyle \int _{- 1}^{0} \left[\right. x \left]\right. d x + \displaystyle \int _{0}^{1} \left[\right. x \left]\right. d x \left]\right.$
$= \frac{1}{2} \left[\right. 1 - 1 \left]\right. - \left[\right. \displaystyle \int _{- 1}^{0} \left(\right. - 1 \left.\right) d x + \displaystyle \int _{0}^{1} 0. d x \left]\right.$
[If -1 $\geq$ x < 0, [x] = -1
If -0 $\geq$ 0 < 1, [x] = -1 ]
$= 0 - [ - x ]_{- 1}^{0} - 0 = 0 - [ - 0 - ( 1 ) ] = 1$