Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. We call $\tilde{f}$ the application from $\mathbf{R}_+$ to $\mathbf{R}$, defined by:
$$\forall x \in \mathbf{R}^+, \tilde{f}(x) = \ln(f(2x))$$
Show that
$$\forall p \in \mathbf{N}^*, \forall x \in \mathbf{R}_+, \tilde{f}(x+p) = \tilde{f}(x) + \sum_{k=0}^{p-1} \ln\left(\frac{2x+2k+1}{2x+2k+2}\right)$$