# How do you integrate int (x+4)e^(-5x) by integration by parts method?

Feb 23, 2017

The answer is $= - \frac{\left(5 x + 21\right)}{25} {e}^{- 5 x} + C$

#### Explanation:

The integration by parts is

$\int u ' v \mathrm{dx} = u v - \int u v ' \mathrm{dx}$

Here, we have

$u ' = {e}^{- 5 x}$, $\implies$, $u = - {e}^{- 5 x} / 5$

$v = x + 4$, $\implies$, $v ' = 1$

Therefore,

$\int \left(x + 4\right) {e}^{- 5 x} \mathrm{dx} = - \frac{x + 4}{5} {e}^{- 5 x} - \int - {e}^{- 5 x} / 5 \mathrm{dx}$

$= - \frac{\left(x + 4\right)}{5} {e}^{- 5 x} - {e}^{- 5 x} / 25 + C$

$= - \frac{\left(5 x + 21\right)}{25} {e}^{- 5 x} + C$