What is the integral of (6-x)e^{x/3} ?

The integral of (6-x)e^{x/3} is -3e^{x/3}x+27e^{x/3}+C

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integral of (6-x)e^{x/3}

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

- 3 e^{\frac{x}{3}} x + 27 e^{\frac{x}{3}} + C

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 6 e^{\frac{x}{3}} d x = 18 e^{\frac{x}{3}}

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

= 18 e^{\frac{x}{3}} - (3 e^{\frac{x}{3}} x - 9 e^{\frac{x}{3}})

Simplify

= - 3 e^{\frac{x}{3}} x + 27 e^{\frac{x}{3}}

Add a constant to the solution

= - 3 e^{\frac{x}{3}} x + 27 e^{\frac{x}{3}} + C

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What is the integral of (6-x)e^{x/3} ?
The integral of (6-x)e^{x/3} is -3e^{x/3}x+27e^{x/3}+C