Integral -- Beta function, Bessel function?

[LagrangeEuler]

Integral
$$\int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3}$$
Is it possible to write integral $\int^{\pi}_0\sin^3xdx$ in form of Beta function, or even Bessel function?

 

[pasmith]

Since $B(\frac12,\frac12) = \pi$ you have $$\int_0^{\pi} \sin^3 x\,dx = \tfrac43B(\tfrac12,\tfrac12) = \tfrac43 \int_0^1 u^{-1/2}(1 - u)^{-1/2}\,du.$$