Question Number 220232 by mnjuly1970 last updated on 09/May/25
$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\:\frac{\pi}{\mathrm{16}}\:<\:\int_{\mathrm{0}} ^{\:\mathrm{1}\:} \sqrt{\frac{{x}\left(\mathrm{1}−{x}\right)}{{sin}\left(\pi{x}\right)+{cos}\left(\pi{x}\right)+\mathrm{2}}}\:{dx}<\frac{\pi}{\mathrm{8}} \\ $$$$\:\:\:\:\: \\ $$
Answered by SdC355 last updated on 10/May/25
$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \:-=\mathrm{0}.\mathrm{245953}……. \\ $$$$\frac{\pi}{\mathrm{16}}=\mathrm{0}.\mathrm{196349540}…… \\ $$$$\frac{\pi}{\mathrm{8}}=\mathrm{0}.\mathrm{39269908}….. \\ $$$$\therefore\frac{\pi}{\mathrm{16}}<\int_{\mathrm{0}} ^{\:\mathrm{1}} \:-<\frac{\pi}{\mathrm{8}} \\ $$