Question Number 131614 by mnjuly1970 last updated on 06/Feb/21
$$\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}\::\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){cos}\left(\mathrm{2}\pi{nx}\right){dx}=\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$$${for}\:{example}\::\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){cos}\left(\mathrm{2}\pi{x}\right){dx}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$