<s>[INST] Prove $B(p, q) = \int \limits_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx$ 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
$$B(p, q) = \int \limits_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx = \int \limits_0^1 \frac{x^{p-1}}{(1+x)^{p+q}}dx + \int \limits_0^1 \frac{x^{q-1}}{(1+x)^{p+q}}dx$$
 [/INST] Note that
$$\begin {align}
2\int_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx &=\int_0^1 \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx+\int_1^\infty \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx\\&=\int_0^\infty \frac{x^{p-1} + x^{q-1}}{(1+x)^{p+q}}dx\\&=\int_0^\infty \frac{x^{p-1} }{(1+x)^{p+q}}dx+\int_0^\infty \frac{x^{q-1}}{(1+x)^{p+q}}dx \\&=B (p,q)+B(q,p)\\&=2 B(p,q)
\end {align}$$
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