Question Number 132110 by Raxreedoroid last updated on 11/Feb/21
$$\int\frac{\mathrm{1}}{\mathrm{ln}\:{x}}{dx}=? \\ $$
Answered by Olaf last updated on 11/Feb/21
$$\mathrm{F}\left({x}\right)\:=\:\int\frac{{dx}}{\mathrm{ln}{x}} \\ $$$$\mathrm{Let}\:{x}\:=\:{e}^{{u}} \\ $$$$\mathrm{F}\left({e}^{{u}} \right)\:=\:\int\frac{{d}\left({e}^{{u}} \right)}{\mathrm{ln}\left({e}^{{u}} \right)}\:=\:\int\frac{{e}^{{u}} }{{u}}{du}\:=\:\mathrm{Ei}\left({u}\right) \\ $$$${u}\:=\:\mathrm{ln}{x} \\ $$$$\mathrm{F}\left({x}\right)\:=\:\mathrm{Ei}\left(\mathrm{ln}{x}\right)\:\left(+\mathrm{C}\right) \\ $$
Commented by Raxreedoroid last updated on 11/Feb/21
$$\mathrm{What}\:\mathrm{is}\:\mathrm{Ei}? \\ $$
Commented by Olaf last updated on 11/Feb/21
$$\mathrm{Ei}\left({x}\right)\:=\:\int_{−\infty} ^{{x}} \frac{{e}^{{t}} }{{t}}{dt} \\ $$$$\left(\mathrm{integral}\:\mathrm{exponential}\:\mathrm{function}\right) \\ $$