Question Number 126710 by slahadjb last updated on 23/Dec/20
$$\int{ln}\left({x}\right){e}^{{x}^{\mathrm{2}} } {dx}\:\:\:\:\:\:??? \\ $$
Answered by Dwaipayan Shikari last updated on 23/Dec/20
$$\int{e}^{{x}^{\mathrm{2}} } {log}\left({x}\right){dx}\:\:\:\:\:{x}^{\mathrm{2}} ={u}\Rightarrow\mathrm{2}{x}=\frac{{du}}{{dx}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{u}^{−\frac{\mathrm{1}}{\mathrm{2}}} {e}^{{u}} {log}\left({u}\right){du} \\ $$$${Generally}\:\int_{\mathrm{0}} ^{{x}} {t}^{{s}−\mathrm{1}} {e}^{−{t}} {dt}\:=\Gamma\left({s},{x}\right)\left(\:\:{Incomplete}\:{Gamma}\right) \\ $$$${Here}\:\frac{\mathrm{1}}{\mathrm{2}}\int{u}^{−\frac{\mathrm{1}}{\mathrm{2}}} {e}^{{u}} {log}\left({u}\right){du}=\frac{\mathrm{1}}{\mathrm{2}}\overset{\bullet} {\Gamma}\left(\frac{\mathrm{1}}{\mathrm{2}},;\right) \\ $$
Commented by slahadjb last updated on 23/Dec/20
$$>{where}\:{is}\:{log}\left({u}\right)\:? \\ $$