Question

Evaluate: ${\int }_0^{10}(x-\left[x\right])dx$ where $[.]$ denotes the greatest integer function.

Answer 1

${\int }_0^{10}(x-[x\left]\right)dx={\int }_0^{10}\left\{x\right\}dx=10{\int }_0^{1}xdx=10{\int }_0^1\frac{x^2}{2}=5(1-0)=5$