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Question Number 132798 by metamorfose last updated on 16/Feb/21
∫_0 ^(π/2) ((√(sin (x)))+(√(cos (x))))dx
$$\overset{\frac{\pi}{\mathrm{2}}} {\int}_{\mathrm{0}} \left(\sqrt{\mathrm{sin}\:\left({x}\right)}+\sqrt{\mathrm{cos}\:\left({x}\right)}\right){dx} \\ $$
Answered by Ñï= last updated on 17/Feb/21
∫_0 ^(π/2) ((√(sinx))+(√(cosx)))dx  =B((3/4),(1/2))  =((Γ((3/4))Γ((1/2)))/(Γ((5/4))))  =4(√π)((Γ((3/4)))/(Γ((1/4))))
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\sqrt{{sinx}}+\sqrt{{cosx}}\right){dx} \\ $$$$={B}\left(\frac{\mathrm{3}}{\mathrm{4}},\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)} \\ $$$$=\mathrm{4}\sqrt{\pi}\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$
Answered by Dwaipayan Shikari last updated on 16/Feb/21
∫_0 ^(π/2) (√(sin(x)))+(√(cos(x))) dx       =((Γ((3/4))Γ((1/2)))/(Γ((5/4))))=((4(√2)π)/(Γ^2 ((1/4))))
$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{sin}\left({x}\right)}+\sqrt{{cos}\left({x}\right)}\:{dx}\:\:\:\:\: \\ $$$$=\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)}=\frac{\mathrm{4}\sqrt{\mathrm{2}}\pi}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\ $$

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