If $n \in \mathbf{N}$, we denote by $D_n$ the improper integral $\int_0^{\pi/2} (\ln(\sin(t)))^n \mathrm{~d}t$.
Verify that if $n \in \mathbf{N}^*$, then
$$(-1)^n D_n = \int_0^{+\infty} \frac{u^n}{\sqrt{\mathrm{e}^{2u} - 1}} \mathrm{~d}u$$
then that
$$D_n \underset{n \to +\infty}{\sim} (-1)^n n!$$