Solution
= Solution
In the integral for $B(q,p)$, put $u=1-x$, so $dx=-du$ and the endpoints are interchanged. Then
$$
B(q,p)=\int_1^0(1-u)^{q-1}u^{p-1}(-du)
=\int_0^1u^{p-1}(1-u)^{q-1}du=\boxed{B(p,q)}.
$$
Thus exchanging the positive parameters leaves the <Euler beta function> unchanged. Positivity ensures convergence at both endpoints of this improper integral.