Question Number 172011 by Mikenice last updated on 23/Jun/22
$${find}\:{integrate}: \\ $$$$\int{x}^{\mathrm{2}} {e}^{{x}} {dx} \\ $$
Answered by puissant last updated on 23/Jun/22
$${P}\:=\:\int{x}^{\mathrm{2}} {e}^{{x}} {dx} \\ $$$$\:\:\:\:\:\begin{cases}{{u}'={e}^{{x}} }\\{{v}={x}^{\mathrm{2}} }\end{cases}\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\:\:\:\begin{cases}{{u}={e}^{{x}} }\\{{v}'=\:\mathrm{2}{x}}\end{cases} \\ $$$$\:{P}=\:{x}^{\mathrm{2}} {e}^{{x}} \:−\:\mathrm{2}\int{xe}^{{x}} {dx} \\ $$$$\:{P}={x}^{\mathrm{2}} {e}^{{x}} \:−\:\mathrm{2}{xe}^{{x}} +\mathrm{2}{e}^{{x}} +{C} \\ $$$$\:{P}\:=\:\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}\right){e}^{{x}} +{C} \\ $$