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Question Number 128373 by BHOOPENDRA last updated on 06/Jan/21

$${\int }_0^1{x}^{\frac{3}{2}}{(1-x)}^{\frac{1}{2}}dx$$

Answered by Dwaipayan Shikari last updated on 06/Jan/21

$${\int }_0^1{x}^{\frac{3}{2}}{(1-x)}^{\frac{1}{2}}dx=\beta (\frac{5}{2},\frac{3}{2})=\frac{\mathrm{\Gamma }\left(\frac{5}{2}\right)\mathrm{\Gamma }\left(\frac{3}{2}\right)}{\mathrm{\Gamma }\left(4\right)}=\frac{3}{8.6}{\mathrm{\Gamma }}^2\left(\frac{1}{2}\right)$$$$=\frac{\sqrt{\pi }.\sqrt{\pi }}{16}=\frac{\pi }{16}$$

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