Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. Determine the domain of definition of $f$ and verify that $$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Determine the domain of definition of $f$ and verify that
$$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$