> [!NOTE] Theorem > Let $\frac{d}{dt} x(t) = f(x(t))g(t)$with $t\in(\alpha,\beta)\subset \mathbb{R},$ $f:\mathbb{R}\to \mathbb{R}$ and $g:(\alpha,\beta)\to \mathbb{R}$ be a [[Separable Differential Equation|separable differential equation]]. > > Then $\int \frac{1}{f(\tilde{x})} \, d\tilde{x} = \int g(\tilde{t}) \, d\tilde{t}$where $\int f(x) \, dx$ denotes an [[Antiderivative|antiderivative]] of $f(x).$ **Proof**: ...