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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int x e^{- \frac{x}{3}} d x

Apply Integration By Parts

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

= \frac{1}{3} (- 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}}

= \frac{1}{3} (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}})

Simplify

\frac{1}{3} (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}) : - e^{- \frac{x}{3}} (x + 3)

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

Add a constant to the solution

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

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