I'm trying to prove that for $p,q>0$, we have $\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$
The hint given suggests that we express $\Gamma(p)\Gamma(q)$ as a double integral, then do a change of variables, but I've been unable thus far to express it as a double integral.
Can anyone get me started or suggest an alternate approach?
Note: This wasn't actually given to me as the $\Gamma$ function, just as a function $f$ satisfying $f(p)=\int_0^\infty e^{-t}t^{p-1}\,dt$ for all $p>0$, but I recognized that. This is in the context of an advanced calculus practice exam.