= Mixed power moments of a beta distribution
{title2=$\mathbb E[X^s(1-X)^t]=B(\alpha+s,\beta+t)/B(\alpha,\beta)$}
For $X\sim\operatorname{Beta}(\alpha,\beta)$ with positive parameters, and real $s>-\alpha$, $t>-\beta$, multiply its density by $x^s(1-x)^t$. The resulting integral is a <beta function> with shifted parameters, hence
$$
\mathbb E[X^s(1-X)^t]=\frac{\Gamma(\alpha+\beta)\Gamma(\alpha+s)\Gamma(\beta+t)}{\Gamma(\alpha)\Gamma(\beta)\Gamma(\alpha+\beta+s+t)}.
$$
The restrictions on the powers are exactly the endpoint-<integrability> conditions. This includes ordinary <moments>, reciprocal <moments> when they exist, and quantities involving both the occupied fraction $X$ and remaining fraction $1-X$. In particular, $s=1,t=0$ gives $\mathbb E[X]=\alpha/(\alpha+\beta)$.<NIST Digital Library of Mathematical Functions/5.12 Beta Function>[NIST DLMF §5.12, equation 5.12.1]{ref}
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