# How do you integrate int xe^(x/2) dx?

Aug 23, 2015

Use integration by parts with $u = x$ and $\mathrm{dv} = {e}^{\frac{x}{2}} \mathrm{dx}$.

#### Explanation:

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

Let $u = x$ $\text{ }$and $\text{ }$ $\mathrm{dv} = {e}^{\frac{x}{2}} \mathrm{dx}$

So $\mathrm{du} = \mathrm{dx}$ $\text{ }$ and $\text{ }$ $v = 2 {e}^{\frac{x}{2}}$ $\text{ }$(use substitution $w = \frac{x}{2}$)

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

$= 2 x {e}^{\frac{x}{2}} - 2 \int {e}^{\frac{x}{2}} \mathrm{dx}$ $\text{ }$(again use substitution $w = \frac{x}{2}$)

$= 2 x {e}^{\frac{x}{2}} - 4 {e}^{\frac{x}{2}} + C$