# How do I find the antiderivative of f(x)=e^(-5x)?

Apr 7, 2018

$\int {e}^{- 5 x} \mathrm{dx} = - \frac{1}{5} {e}^{- 5 x} + C$

#### Explanation:

In general, the antiderivative of an exponential in the form

$\int {e}^{a x} \mathrm{dx} = \frac{1}{a} {e}^{a x} + C$ where $a$ is some non-zero constant.

This makes sense -- were we to differentiate $\frac{1}{a} {e}^{a x} + C ,$ we'd get $\frac{a}{a} {e}^{a x} = {e}^{a x}$, making this a suitable antiderivative.

Thus,

$\int {e}^{- 5 x} \mathrm{dx} = - \frac{1}{5} {e}^{- 5 x} + C$