Question Number 163002 by mnjuly1970 last updated on 03/Jan/22
$$ \\ $$$$\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:{i}:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\:\right)^{\:{n}} }{\left({n}\:+\frac{\mathrm{1}}{\mathrm{2}}\right){cosh}\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\pi}\:=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:{ii}:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{sin}\left(\:\pi\:{x}\:\right)}{{x}^{\:{x}} \left(\:\mathrm{1}−{x}\:\right)^{\:\mathrm{1}−{x}} }\:\frac{{dx}}{\mathrm{1}+{x}}\:=\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:−−− \\ $$