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(v) \( \int_{0}^{\frac{\pi}{2}} \sin ^{2} x \cos ^{4} x d x \)

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\(\int_0^{\frac{\pi}{2}} sin^2x\,cos^4x\,dx\)

\(\frac{\Gamma(\frac{2+1}{2})\Gamma(\frac{4+1}{2})}{2\Gamma{(\frac{2+4+2}{2})}}\)

(∵ \(\int_0^{\frac{\pi}{2}} sin^m\theta\,cos^n\theta\,d\theta\) = \(\frac{\Gamma(\frac{m+1}{2})\Gamma(\frac{n+1}{2})}{2\Gamma{(\frac{m+n+2}{2})}}\))

\(\frac{\Gamma(\frac{3}{2})\Gamma(\frac{5}{2})}{2\Gamma{(4)}}\)

\(\frac{\frac{1}{2}\Gamma(\frac{1}{2})\times \frac{3}{2}\times \frac{1}{2}\Gamma(\frac{1}{2})}{2\times 3!}\)

(∵ \(\Gamma(n+1)\) = 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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