Compute some integrals of step functions

Let $n \in \mathbb{Z}_{>0}$ , prove
$\int_0^n [t] \, dt = \frac{n(n-1)}{2}.$
Let $x \in \mathbb{R}_{>0}$ , and define
$f(x) = \int_0^x [t] \, dt.$

Draw the graph of $f$ for $0 \leq x \leq 4$ .

Proof. Let $P = \{ 0,1,2, \ldots, n \}$ be a partition of $[0,n]$ . Then, by the definition of the greatest integer function, $[t]$ is constant on the open subintervals of $P$ , so
$\begin{align*} \int_0^n [t] \, dt &= \sum_{k=1}^n (k-1)(k-(k-1)) & (s_k = k-1 \text{ since } [t]=k-1 \text{ if } k-1 < t < k)\\ &= \sum_{k=1}^n (k-1) \\ &= \sum_{k=0}^{n-1} k \\ &= \frac{n(n-1)}{2}. \end{align*}$

The final equality follows from here (I.4.7, #5) $. \qquad \blacksquare$
The graph is: