\(\int_0^\frac{\pi}{2} sin^3\theta\,cos^5\theta\,{d}\theta\)
= \(\frac{\Gamma(\frac{3+1}{2})\Gamma(\frac{5+1}{2})}{2\Gamma(\frac{3+5+2}{2})}\)
= \(\frac{\Gamma(2)\Gamma(3)}{2\Gamma(5)}\)
= \(\frac{1!2!}{2\times 4!}\)
(∵ \(\Gamma\)(n+1) = n!, n ∈ N)
= \(\frac{1}{24}\)
Hence,
\(\int_0^\frac{\pi}{2} sin^3\theta\,cos^5\theta\,{d}\theta\) = \(\frac{1}{24}\)
