What is the integral of xe^{(-x)/3} ?

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

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integral of xe^{(-x)/3}

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

- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}} + C

Solution steps

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

Simplify

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

Apply Integration By Parts

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

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

= - 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}

Add a constant to the solution

= - 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}} + C

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