TITLE: Prove $B(p, q) = \int \limits_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx$
QUESTION [2 upvotes]: Prove the equality $$B(p, q) = \int \limits_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx$$ for $p > 0, q > 0$
I think the first step is
$$B(p, q) = \int \limits_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx = \int \limits_0^1 \frac{x^{p-1}}{(1+x)^{p+q}}dx + \int \limits_0^1 \frac{x^{q-1}}{(1+x)^{p+q}}dx$$

REPLY [0 votes]: Note that
$$\begin {align}
2\int_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx &=\int_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx+\int_1^\infty \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx\\&=\int_0^\infty \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx\\&=\int_0^\infty \frac{x^{p-1} }{(1+x)^{p+q}}dx+\int_0^\infty \frac{x^{q-1}}{(1+x)^{p+q}}dx \\&=B (p,q)+B(q,p)\\&=2 B(p,q)
\end {align}$$
{"set_name": "stack_exchange", "score": 2, "question_id": 2160680}