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(vii) \( \int_{0}^{\frac{\pi}{2}} \sin ^{3} \theta \cos ^{5} \theta d \theta \)

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\(\int_0^\frac{\pi}{2} sin^3\theta\,cos^5\theta\,{d}\theta\)

\(\frac{\Gamma(\frac{3+1}{2})\Gamma(\frac{5+1}{2})}{2\Gamma(\frac{3+5+2}{2})}\) 

\(\frac{\Gamma(2)\Gamma(3)}{2\Gamma(5)}\)

\(\frac{1!2!}{2\times 4!}\)

(∵ \(\Gamma\)(n+1) = n!, n ∈ N)

\(\frac{1}{24}\)

Hence,

\(\int_0^\frac{\pi}{2} sin^3\theta\,cos^5\theta\,{d}\theta\) = \(\frac{1}{24}\) 

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