# How do you integrate int re^(r/2) by integration by parts method?

Oct 20, 2016

$\int r {e}^{\frac{r}{2}} \mathrm{dr} = \left(2 r - 4\right) {e}^{\frac{r}{2}} + C$

#### Explanation:

Let $u = r$ then $u ' = 1$
and $v ' = {e}^{\frac{r}{2}}$ then $v = 2 {e}^{\frac{r}{2}}$
$\int u v ' = u v - \int u ' v$
so $\int r {e}^{\frac{r}{2}} \mathrm{dr} = 2 r {e}^{\frac{r}{2}} - \int 2 {e}^{\frac{r}{2}} \mathrm{dr}$
$= 2 r {e}^{\frac{r}{2}} - 2 \cdot 2 {e}^{\frac{r}{2}}$
$= 2 r {e}^{\frac{r}{2}} - 4 {e}^{\frac{r}{2}} + C$
$= \left(2 r - 4\right) {e}^{\frac{r}{2}} + C$