Question

Evaluate ${\int }_{-1}^1(x-[x\left]\right)dx$; (where $[.]$ denotes the greatest integral part of $x$.)

Answer 1

$I={\int }_{-1}^1(x-[x\left]\right)dx={\int }_{-1}^{1}x.dx-{\int }_{-1}^1\left[x\right]dx$
$={\left(\frac{x^2}{2}\right)}_{-1}^1-({\int }_{-1}^{0}x.dx+{\int }_0^1\left[x\right]dx)$
$=\frac{1}{2}(1-1)-({\int }_{-1}^0-1dx+{\int }_0^{1}0dx)$
$=0+{\left(x\right)}_{-1}^0-0=1$