If $n \in \mathbf{N}$, we denote by $D_n$ the improper integral $\int_0^{\pi/2} (\ln(\sin(t)))^n \mathrm{~d}t$.
Justify that, if $n \in \mathbf{N}$, the improper integral $D_n$ is convergent, then show that
$$D_1 = \int_0^{\pi/2} \ln(\cos(t)) \mathrm{d}t$$