Question Number 217794 by yamane last updated on 21/Mar/25
$${I}\:{need}\:{help} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {e}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:{dx}\:\:\left({n}\in\mathbb{N}\right) \\ $$
Answered by Wuji last updated on 21/Mar/25
$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {e}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}{dx}\:\:\:\left({n}\in\mathbb{N}\right) \\ $$$${e}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} −\frac{{x}^{\mathrm{2}} }{\mathrm{2}}{dx}\:\:\Rightarrow{e}\left(\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}} {dx}\right)−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {dx} \\ $$$${e}\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)−\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$${e}\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)−\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$\frac{{e}}{{n}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{6}} \\ $$