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Question
\[\int x^2 e^{- x} \text{ dx }\]
Sum
Solution
\[\int x^2 e^{- x} \text{ dx }\]
` " Taking x"^2" as the first function and e"^- x " as the second function ".`
\[ = x^2 \int e^{- x} dx - \int\left( \frac{d}{dx} x^2 \int e^{- x} dx \right)dx\]
\[ = - x^2 e^{- x} - \int2x\left( e^{- x} \right)\left( - 1 \right)dx\]
\[ = - x^2 e^{- x} + 2\int x e^{- x} dx\]
\[ = - x^2 e^{- x} + 2\left[ - x e^{- x} + \int e^{- x} dx \right]\]
\[ = - x^2 e^{- x} + 2\left[ - x e^{- x} - e^{- x} \right] + C\]
\[ = - e^{- x} \left[ x^2 + 2x + 2 \right] + C\]
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