Question Number 196832 by Frix last updated on 01/Sep/23
$$\int{x}\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{2}{x}}} {dx}=? \\ $$
Commented by mokys last updated on 02/Sep/23
$${u}\:=\:\frac{\mathrm{1}}{\mathrm{2}{x}}\:\rightarrow\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}{u}}\:\rightarrow\:{dx}\:=\:−\:\frac{{du}}{\mathrm{2}{u}^{\mathrm{2}} } \\ $$$$ \\ $$$$\:\int\:{x}\:{e}^{\frac{\mathrm{1}}{\mathrm{2}{x}}} {dx}\:=\:−\frac{\mathrm{1}}{\mathrm{4}}\int\:\frac{{e}^{{u}} }{{u}^{\mathrm{3}} }\:{du}\:=\:−\:\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\mathrm{2}}\:{Ei}\left({u}\right)\:−\frac{{e}^{{u}} }{\mathrm{2}{u}}\:−\:\frac{{e}^{{u}} }{\mathrm{2}{u}^{\mathrm{2}} }\right)\:+\:{K}\:\: \\ $$$$ \\ $$$$\: \\ $$