We denote $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$, and for all $p \in \mathbb{Z}$,
$$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$
Show that $I_{0} = 1$ and that, for all $p \in \mathbb{N}^{*}$, $I_{p} = 0$.