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Integration Questions

Question Number 132925 by metamorfose last updated on 17/Feb/21

$$\left(\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{sin}\left({x}\right)}{dx}\right)^{\mathrm{2}} +\left(\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{cos}\left({x}\right)}{dx}\right)^{\mathrm{2}} <\mathrm{8}\left(\sqrt{\mathrm{2}}−\mathrm{1}\right) \\$$

Answered by Dwaipayan Shikari last updated on 17/Feb/21

$$\Phi=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{sin}\left({x}\right)}\:{dx}=\mathrm{2}\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}=\mathrm{2}\sqrt{\mathrm{2}}\pi\frac{\mathrm{1}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\$$ $$\Psi=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{cos}\left({x}\right)}{dx}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{sin}\left({x}\right)}{dx}=\mathrm{2}\sqrt{\mathrm{2}}\pi.\frac{\mathrm{1}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\$$ $$\Phi^{\mathrm{2}} +\Psi^{\mathrm{2}} =\frac{\mathrm{16}\pi^{\mathrm{2}} }{\Gamma^{\mathrm{4}} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)} \\$$

Commented byÑï= last updated on 17/Feb/21

$$\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\sqrt{\pi}....? \\$$ $$\Gamma\left(\mathrm{1}−{x}\right)\Gamma\left({x}\right)=\frac{\pi}{{sin}\pi{x}}...? \\$$ $${Confused}\:{why}\:{you}\:{got}\:{that}\:{result}. \\$$