What is the integral of x^2e^{-x/3} ?

The integral of x^2e^{-x/3} is -3e^{-x/3}x^2+6(-3e^{-x/3}x-9e^{-x/3})+C

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integral of x^2e^{-x/3}

\int x^{2} e^{- \frac{x}{3}} d x

- 3 e^{- \frac{x}{3}} x^{2} + 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}) + C

Solution steps

\int x^{2} e^{- \frac{x}{3}} d x

Apply Integration By Parts

= - 3 e^{- \frac{x}{3}} x^{2} - \int - 6 e^{- \frac{x}{3}} x d x

\int - 6 e^{- \frac{x}{3}} x d x = - 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}})

= - 3 e^{- \frac{x}{3}} x^{2} - (- 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}))

Simplify

= - 3 e^{- \frac{x}{3}} x^{2} + 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}})

Add a constant to the solution

= - 3 e^{- \frac{x}{3}} x^{2} + 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}) + C

What is the integral of x^2e^{-x/3} ?
The integral of x^2e^{-x/3} is -3e^{-x/3}x^2+6(-3e^{-x/3}x-9e^{-x/3})+C