Evaluate the following integrals: ^2e^ -x dx - Vidyadip

Question

Evaluate the following integrals:$\int\text{x}^2\text{e}^{-\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\text{x}^2\text{e}^{-\text{x}}\text{dx}$
Using integration by parts,
$\text{I}=\text{x}^2\int\text{e}^{-\text{x}}\text{dx}-\int(2\text{x}\int\text{e}^{-\text{x}}\text{dx})$
$=-\text{x}^2\text{e}^{-\text{x}}-\int(2\text{x})(-\text{e}^{-\text{x}})$
$=-\text{x}^2\text{e}^{-\text{x}}+2\int\text{xe}^{-\text{x}}\text{dx}$
$=-\text{x}^2\text{e}^{-\text{x}}+2[\text{x}\int\text{e}^{-\text{x}}\text{dx}-\int(1\times\int\text{e}^{-\text{x}}\text{dx})\text{dx}]$
$=-\text{x}^2\text{e}^{-\text{x}}+2[\text{x}(-\text{e}^{-\text{x}})-\int(-\text{e}^{-\text{x}})\text{dx}]$
$=-\text{x}^2\text{e}^{-\text{x}}-2\text{xe}^{-\text{x}}+2\int\text{e}^{-\text{x}}\text{dx}$
$\text{I}=-\text{x}^2\text{e}^{-\text{x}}-2\text{xe}^{-\text{x}}-2\text{e}^{-\text{x}}+\text{C}$
$\text{I}=-\text{e}^{-\text{x}}(\text{x}^2+2\text{x}+2)+\text{C}$

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