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(v) \( \int_{0}^{\frac{\pi}{2}} \sin ^{2} x \cos ^{4} x d x \) |
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Answer» \(\int_0^{\frac{\pi}{2}} sin^2x\,cos^4x\,dx\) = \(\frac{\Gamma(\frac{2+1}{2})\Gamma(\frac{4+1}{2})}{2\Gamma{(\frac{2+4+2}{2})}}\) (∵ \(\int_0^{\frac{\pi}{2}} sin^m\theta\,cos^n\theta\,d\theta\) = \(\frac{\Gamma(\frac{m+1}{2})\Gamma(\frac{n+1}{2})}{2\Gamma{(\frac{m+n+2}{2})}}\)) = \(\frac{\Gamma(\frac{3}{2})\Gamma(\frac{5}{2})}{2\Gamma{(4)}}\) = \(\frac{\frac{1}{2}\Gamma(\frac{1}{2})\times \frac{3}{2}\times \frac{1}{2}\Gamma(\frac{1}{2})}{2\times 3!}\) (∵ \(\Gamma(n+1)\) = n\(\Gamma\)(n),n>0 & \(\Gamma\)(n+1) = n!, n∈N) = \(\frac{3}{8}\) x \(\frac{\sqrt{\pi}\times {\sqrt\pi}}{2\times 6}\) = \(\frac{\pi}{32}\). Hence, \(\int_0^{\frac{\pi}{2}} sin^2x\,cos^4x\,dx\) = \(\frac{\pi}{32}\). |
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