What is the integral of (7-x)e^{x/2} ?

The integral of (7-x)e^{x/2} is -2e^{x/2}x+18e^{x/2}+C

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integral of (7-x)e^{x/2}

\int (7 - x) e^{\frac{x}{2}} d x

- 2 e^{\frac{x}{2}} x + 18 e^{\frac{x}{2}} + C

Solution steps

\int (7 - x) e^{\frac{x}{2}} d x

Expand

(7 - x) e^{\frac{x}{2}} : 7 e^{\frac{x}{2}} - e^{\frac{x}{2}} x

= \int 7 e^{\frac{x}{2}} - e^{\frac{x}{2}} 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 7 e^{\frac{x}{2}} d x - \int e^{\frac{x}{2}} x d x

\int 7 e^{\frac{x}{2}} d x = 14 e^{\frac{x}{2}}

\int e^{\frac{x}{2}} x d x = 4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

= 14 e^{\frac{x}{2}} - 4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

Simplify

14 e^{\frac{x}{2}} - 4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}) : - 2 e^{\frac{x}{2}} x + 18 e^{\frac{x}{2}}

= - 2 e^{\frac{x}{2}} x + 18 e^{\frac{x}{2}}

Add a constant to the solution

= - 2 e^{\frac{x}{2}} x + 18 e^{\frac{x}{2}} + C

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What is the integral of (7-x)e^{x/2} ?
The integral of (7-x)e^{x/2} is -2e^{x/2}x+18e^{x/2}+C