demostration of interger part integration. I need help for solving this demostration, I  appreciate your suggestions very much.

$$\begin{array}{rclr}
\int ^{n}_{0}[x] dx= \frac{n(n-1)}{2} 
\end{array}$$

Pd. If you have any suggestion of a book that deepens on the subject please communicate it.
Hint: $\lfloor x \rfloor = k$ for $x \in [k,k+1)$ for $k = 0,1,2,\ldots,n-1$. Hence
$$\int_0^n \lfloor x \rfloor \; dx = \sum_{k=0}^{n-1} \int_{k}^{k+1} \lfloor x \rfloor \; dx = \sum_{k=0}^{n-1} \int_{k}^{k+1} k \; dx,$$
where the integrand is a constant with respect to $x$.
For any positive integer n,  $[x]=n$ for $n\le x&lt; n+1$
$$\int_0^n[x]dx=0+1+2+3+4+\ldots +(n-1)=\frac{n(n-1)}{2}$$