Question Number 132798 by metamorfose last updated on 16/Feb/21
$$\overset{\frac{\pi}{\mathrm{2}}} {\int}_{\mathrm{0}} \left(\sqrt{\mathrm{sin}\:\left({x}\right)}+\sqrt{\mathrm{cos}\:\left({x}\right)}\right){dx} \\ $$
Answered by Ñï= last updated on 17/Feb/21
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\sqrt{{sinx}}+\sqrt{{cosx}}\right){dx} \\ $$$$={B}\left(\frac{\mathrm{3}}{\mathrm{4}},\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)} \\ $$$$=\mathrm{4}\sqrt{\pi}\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$
Answered by Dwaipayan Shikari last updated on 16/Feb/21
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{sin}\left({x}\right)}+\sqrt{{cos}\left({x}\right)}\:{dx}\:\:\:\:\: \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)}=\frac{\mathrm{4}\sqrt{\mathrm{2}}\pi}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$