= Solution
Start from the product of two <Gamma function> integrals:
$$
\Gamma(p)\Gamma(q)=
\int_0^\infty\int_0^\infty e^{-(x+y)}x^{p-1}y^{q-1}\,dx\,dy.
$$
Use the <sum-and-ratio substitution for gamma integrals>
$$
r=x+y,\qquad t=\frac{x}{x+y},\qquad
x=rt,\quad y=r(1-t),
$$
whose Jacobian is $r$. Absolute convergence for $\operatorname{Re}p,\operatorname{Re}q>0$ justifies the change of variables and separates the integral:
$$
\Gamma(p)\Gamma(q)
=\left(\int_0^\infty e^{-r}r^{p+q-1}\,dr\right)
\left(\int_0^1t^{p-1}(1-t)^{q-1}\,dt\right)
=\Gamma(p+q)B(p,q).
$$
Therefore
$$
\boxed{\ B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}\ }.
$$
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