Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. Show that for every natural number $n$, $$f(n)f(n+1) = \frac{\pi}{2(n+1)}$$ then that: $$f(x) \underset{x \to +\infty}{\sim} \sqrt{\frac{\pi}{2x}}$$
Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Show that for every natural number $n$,
$$f(n)f(n+1) = \frac{\pi}{2(n+1)}$$
then that:
$$f(x) \underset{x \to +\infty}{\sim} \sqrt{\frac{\pi}{2x}}$$