Are $\int_{C}^{} f(x,y) ds $ and $\int_{C}^{} F(x,y) = \int_{C}^{} P(x,y) dx + \int_{C}^{} Q(x,y) dy$ equivalent to each other for some curve C? If not, when are they not interchangeable? And if so, when is there an advantage to using one over another?
Edit: it seems that the difference is when the function being integrated has $x$ and $y$ being multiplied or divided, i.e, inseparable into two separate functions. Is this correct?
Strictly speaking there is no real diffetence between ds, dx, and dy. Just that we integrate over a different variable.
However, s is commonly used as a path length parameter. And x and y are commonly used to integrate parallel to the x-axis respectively the y-axis.
In your case the first integral assumes that we can write x and y as functions of s. That is, we have x(s) and y(s) that parametrize the curve C.
The sevond integral integrates along the first parameter of P, which is x, which represents the coordinate along the x axis. It means that we consider y a function of x, that is y(x) so that it follows the curve C.
As an example consider $(x(s), y(s))=(\cos s,\sin s)$ and $(x,y(x))=(x,\sqrt{x^2+y^2})$ respectively $(x(y), y)=(\sqrt{x^2+y^2},y)$. These are 3 different parametrizations of the first quarter of the unit circle.