We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Deduce that, for all $x > 0$, $$J_{n}(x) = \frac{n!}{x(x+1) \cdots (x+n-1)(x+n)}$$
We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by:
$$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$
Deduce that, for all $x > 0$,
$$J_{n}(x) = \frac{n!}{x(x+1) \cdots (x+n-1)(x+n)}$$