What is the integral of x^2e^{(-x)/2} ?

The integral of x^2e^{(-x)/2} is -2e^{-x/2}x^2+4(-2e^{-x/2}x-4e^{-x/2})+C

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integral of x^2e^{(-x)/2}

\int x^{2} e^{\frac{- x}{2}} d x

- 2 e^{- \frac{x}{2}} x^{2} + 4 (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}}) + C

Solution steps

\int x^{2} e^{\frac{- x}{2}} d x

Apply Integration By Parts

: - 2 e^{- \frac{x}{2}} x^{2} - \int - 4 e^{- \frac{x}{2}} x d x

= - 2 e^{- \frac{x}{2}} x^{2} - \int - 4 e^{- \frac{x}{2}} x d x

\int - 4 e^{- \frac{x}{2}} x d x = - 4 (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}})

= - 2 e^{- \frac{x}{2}} x^{2} - (- 4 (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}}))

Simplify

= - 2 e^{- \frac{x}{2}} x^{2} + 4 (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}})

Add a constant to the solution

= - 2 e^{- \frac{x}{2}} x^{2} + 4 (- 2 e^{- \frac{x}{2}} x - 4 e^{- \frac{x}{2}}) + C

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What is the integral of x^2e^{(-x)/2} ?
The integral of x^2e^{(-x)/2} is -2e^{-x/2}x^2+4(-2e^{-x/2}x-4e^{-x/2})+C