Question

Evaluate $\underset{0}{\overset{m}{\int }}\left[x\right]dx$

Answer 1

${\int }_0^m\left[x\right]dx$

Greatest integer function gives $x-1$ on non integer and gives $x$ on integers. So integration of only some integer value is very much less than the total integration

So this can be opened as –

$={\int }_0^{m}x-1dx$

$=[\frac{x^2}{2}-x]{\int }_0^m$

$=\frac{m^2}{2}-m$