# How do you integrate int xe^(x/2) using integration by parts?

Apr 20, 2018

$\int x {e}^{\frac{x}{2}} \mathrm{dx} = 2 {e}^{\frac{x}{2}} \left(x - 2\right) + C$

#### Explanation:

Make the following selections:

$u = x$

$\mathrm{du} = \mathrm{dx}$

$\mathrm{dv} = {e}^{\frac{x}{2}} \mathrm{dx}$

$v = \int {e}^{\frac{x}{2}} \mathrm{dx} = 2 {e}^{\frac{x}{2}}$

Then

$u v - \int v \mathrm{du} = 2 x {e}^{\frac{x}{2}} - 2 \int {e}^{\frac{x}{2}} \mathrm{dx} = 2 x {e}^{\frac{x}{2}} - 4 {e}^{\frac{x}{2}} + C$

Factoring out the exponential, we obtain

$\int x {e}^{\frac{x}{2}} \mathrm{dx} = 2 {e}^{\frac{x}{2}} \left(x - 2\right) + C$