Upon reading some mathematical literature, I have encountered the following computation:

$x\in X$, a Banach space, $\alpha=\text{Re }(z)$ for $z\in\mathbb{C}$ and $\omega$ is the growth bound.

$$\int\_{0}^{\infty}e^{-(\alpha-\omega)t}\|x\|dt=\frac{1}{\alpha-\omega}\|x\|$$

Could someone please explain this to me?