# What is the integral of f(x)=xe^(2x)?

Jan 29, 2015

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

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

In your case you can choose:
$f \left(x\right) = {e}^{2 x}$
$g \left(x\right) = x$

$\int x {e}^{2 x} \mathrm{dx} = x {e}^{2 x} / 2 - \int 1 \cdot {e}^{2 x} / 2 \mathrm{dx} =$
$= x {e}^{2 x} / 2 - {e}^{2 x} / 4 + c$

Hope it helps