Let for $\alpha\in \mathbb R$ $$\Gamma(\alpha):= \int_{0}^{\infty}x^{\alpha-1}e^{-x}dx.$$
Recently I bumped into the following: $$\int_{0}^{1}x^{u}(1-x)^{v}dx=\frac{\Gamma(u+1)\Gamma(v+1)}{\Gamma(u+v+2)}.$$
The identity seems to be very common, but I wonder how to prove it (in as simple, as possible low-level way).
