This has to be a really stupid question, but here it goes:
$$ \int_0^1 t^{x-1} (1 -t)^{y-1} dt= \int_0^1 t^{x-1} dt- \int_0^1 t^{x+y-2} dt= $$
$$ \frac{t^x}{x} |^1_0 + \frac{t^{x+y-1}}{x+y-1} |^1_0 = \frac{1}{x} + \frac{1}{x+y-1} $$
Something extremely obvious has to be going on, since I never see this closed form, but I don't see what exactly.
What is wrong??
This:
$$\large \boxed{t^{x-1}(1-t)^{y-1} \color{red}{\neq} t^{x-1} - t^{x-1}\cdot t^{y-1}}$$