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Question Number 141388 by cesarL last updated on 18/May/21

$${\int }_0^{\pi /2}\sqrt{(senx\cdot cosx)}dx$$$$Help$$

Answered by Dwaipayan Shikari last updated on 18/May/21

$${\int }_0^{\frac{\pi }{2}}{sin}^{2\alpha -1}\theta {cos}^{2\alpha -1}\theta d\theta =\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)}{2\mathrm{\Gamma }(\alpha +\beta )}$$$${\int }_0^{\frac{\pi }{2}}{sin}^{2\left(\frac{3}{4}\right)-1}\theta {cos}^{2\left(\frac{3}{4}\right)-1}d\theta =\frac{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}{2\mathrm{\Gamma }\left(\frac{3}{2}\right)}=\frac{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}{\sqrt{\pi }}$$

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