Question Number 196515 by hardmath last updated on 26/Aug/23
$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:+\infty} \:\mathrm{x}^{\boldsymbol{\pi}} \:\mathrm{e}^{−\boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:=\:? \\ $$
Answered by aleks041103 last updated on 26/Aug/23
$$\int_{\mathrm{0}} ^{\infty} {x}^{{s}} {e}^{−{x}} {dx}=\Gamma\left({s}+\mathrm{1}\right) \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\:\infty} {x}^{\pi} {e}^{−{x}} {dx}\:=\:\Gamma\left(\pi+\mathrm{1}\right) \\ $$
Commented by aleks041103 last updated on 26/Aug/23
$${No}\:{normal}\:{answer} \\ $$
Commented by hardmath last updated on 26/Aug/23
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$