We fix an integer $n \in \mathbb{N}^*$ and consider a function $g : [0,n] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$. The polynomials $B_m$ are defined by $B_m(x) = \sum_{k=0}^m \binom{m}{k} b_k x^{m-k}$.
Deduce that for all integer $m \geqslant 2$,
$$\int_0^n g(x)\,\mathrm{d}x = \sum_{k=0}^{n-1} \frac{g(k)+g(k+1)}{2} + \sum_{p=2}^m \frac{(-1)^{p-1} b_p}{p!}\left(g^{(p-1)}(n) - g^{(p-1)}(0)\right) + \frac{(-1)^m}{m!} \int_0^n B_m(x - \lfloor x \rfloor) g^{(m)}(x)\,\mathrm{d}x.$$