Question Number 134716 by metamorfose last updated on 06/Mar/21
$$\int\frac{{ln}\left({x}\right)}{{x}−\mathrm{1}}{dx}=…?? \\ $$
Answered by Lordose last updated on 06/Mar/21
$$\int\frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}−\mathrm{1}}\mathrm{dx}\:\overset{\mathrm{u}=\mathrm{x}−\mathrm{1}} {=}\int\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{u}\right)}{\mathrm{u}}\mathrm{du}\:=\:−\mathrm{Li}_{\mathrm{2}} \left(−\mathrm{u}\right)\:+\:\mathrm{C} \\ $$$$\Omega\:=\:−\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}−\mathrm{x}\right)\:+\:\mathrm{C} \\ $$