# How do I evaluate intz/e^(3z) dz?

Jan 29, 2015

I would first write your integral in a more friendly form:
$\int \frac{z}{{e}^{3 z}} \mathrm{dz} = \int z \cdot {e}^{- 3 z} \mathrm{dz}$

I would then use integration by parts:
Where you have:
$\int f \left(z\right) \cdot g \left(z\right) \mathrm{dz} = F \left(z\right) \cdot g \left(z\right) - \int F \left(z\right) \cdot g ' \left(z\right) \mathrm{dz}$

Where:
$F \left(z\right) = \int f \left(z\right) \mathrm{dz}$
$g ' \left(z\right)$ is the derivative of $g \left(z\right)$

In your case you can choose:
$f \left(z\right) = {e}^{- 3 z}$
$g \left(z\right) = z$
And:
$\int z \cdot {e}^{- 3 z} \mathrm{dz} = z \cdot {e}^{- 3 z} / \left(- 3\right) - \int 1 \cdot {e}^{- 3 z} / \left(- 3\right) \mathrm{dz} =$
$= z \cdot {e}^{- 3 z} / \left(- 3\right) - {e}^{- 3 z} / 9 + c$