= Trigonometric form of the beta function
{title2=$B(p,q)=2\int_0^{\pi/2}\sin^{2p-1}\theta\cos^{2q-1}\theta\,d\theta$}
For positive real $p,q$, substitute $x=\sin^2\theta$ in the defining <Euler beta function> integral. The increasing substitution maps $[0,\pi/2]$ onto $[0,1]$ and has $dx=2\sin\theta\cos\theta\,d\theta$. Combining this factor with $x^{p-1}(1-x)^{q-1}$ gives the displayed identity. Positivity of both parameters ensures convergence at the two endpoints, even when the integrand is unbounded there.
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