Question

Knowing that \[ ln(x) := \int_1^x{\frac{1}{t}dt} \]
How is \[ ln(\phi(x)) = \int{\frac{\frac{d\phi}{dx}(x)}{\phi(x)}dx} \]

Answer 1

we know that,
$$\begin{align*}\frac{d\,\ln(\phi(x))}{dx}&=\frac{d\,\ln(\phi(x))}{d\,[\phi(x)]}.\frac{d\,[\phi(x)]}{dx}\\
&=\frac{1}{\phi(x)}\frac{d\,[\phi(x)]}{dx}\end{align*}$$
integrating both sides wrt x,

$$\ln(\phi(x))=\int \frac{\frac{d\,[\phi(x)]}{dx}}{\phi(x)}dx$$