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(vii) \( \int_{0}^{\frac{\pi}{2}} \sin ^{3} \theta \cos ^{5} \theta d \theta \) |
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Answer» \(\int_0^\frac{\pi}{2} sin^3\theta\,cos^5\theta\,{d}\theta\) = \(\frac{\Gamma(\frac{3+1}{2})\Gamma(\frac{5+1}{2})}{2\Gamma(\frac{3+5+2}{2})}\) = \(\frac{\Gamma(2)\Gamma(3)}{2\Gamma(5)}\) = \(\frac{1!2!}{2\times 4!}\) (∵ \(\Gamma\)(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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