Q:

Integration of Stieltjes integral

I am looking for a motivation behind the following operation
$\int \frac{1}{y(t)}dy(t)=ln(\mid y(t) \mid)+c$

A:

If $y(t)$ is derivable, then by chain rule:
$$\int{\frac{dy(t)}{y(t)}}=\int{\frac{y'(t)}{y(t)}dt}=\int{\frac{d\,\ln(y(t))}{dt}dt}=\ln{(|y(t)|)}+C$$

