What is the integral of 1/3 xe^{x/6} ?

The integral of 1/3 xe^{x/6} is 2e^{x/6}(x-6)+C

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integral of 1/3 xe^{x/6}

\int \frac{1}{3} x e^{\frac{x}{6}} d x

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

Solution steps

\int \frac{1}{3} x e^{\frac{x}{6}} 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}{6}} d x

Apply Integration By Parts

: 6 e^{\frac{x}{6}} x - \int 6 e^{\frac{x}{6}} d x

= \frac{1}{3} (6 e^{\frac{x}{6}} x - \int 6 e^{\frac{x}{6}} d x)

\int 6 e^{\frac{x}{6}} d x = 36 e^{\frac{x}{6}}

= \frac{1}{3} (6 e^{\frac{x}{6}} x - 36 e^{\frac{x}{6}})

Simplify

\frac{1}{3} (6 e^{\frac{x}{6}} x - 36 e^{\frac{x}{6}}) : 2 e^{\frac{x}{6}} (x - 6)

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

Add a constant to the solution

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

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What is the integral of 1/3 xe^{x/6} ?
The integral of 1/3 xe^{x/6} is 2e^{x/6}(x-6)+C