Question Number 131614 by mnjuly1970 last updated on 06/Feb/21
![...nice calculus... prove that : ∫_0 ^( 1) ln(Γ(x))cos(2πnx)dx=(1/(4n)) for example :∫_0 ^( 1) ln(Γ(x))cos(2πx)dx=(1/4)](https://www.tinkutara.com/question/Q131614.png)
$$\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}\::\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){cos}\left(\mathrm{2}\pi{nx}\right){dx}=\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$$${for}\:{example}\::\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){cos}\left(\mathrm{2}\pi{x}\right){dx}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$