Question Number 160276 by amin96 last updated on 27/Nov/21
$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)}{\mathrm{1}−\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}}=? \\ $$
Answered by mnjuly1970 last updated on 27/Nov/21
![∫_0 ^( 1) (( ln(x).ln(1−x))/x)dx= [−li_2 (x).ln(x)]_0 ^1 + ∫_0 ^( 1) ((li_2 (x))/x)dx= li_3 (1) = ζ (3)](https://www.tinkutara.com/question/Q160280.png)
$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}\left({x}\right).{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx}=\:\left[−{li}_{\mathrm{2}} \left({x}\right).{ln}\left({x}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$\:\:\:+\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{li}_{\mathrm{2}} \left({x}\right)}{{x}}{dx}=\:{li}_{\mathrm{3}} \left(\mathrm{1}\right)\:=\:\zeta\:\left(\mathrm{3}\right) \\ $$