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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.

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    how have you aparted the term $$t^{x-1}(1-t)^{y-1}$$? this is impossible – 2017-01-22
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    In general $(a+b)^{z}\neq a^z+b^z$. – 2017-01-22

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What is wrong??

This:

$$\large \boxed{t^{x-1}(1-t)^{y-1} \color{red}{\neq} t^{x-1} - t^{x-1}\cdot t^{y-1}}$$

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    of course. God, I was blind. – 2017-01-22
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    @elelias lol!$~~~$ – 2017-01-22
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    I swear some days... – 2017-01-22