Given a path C and a force field F, the work done on the particle can be found by
$$ \int_C \vec{F}\cdot \vec{r} \,dr $$
This seems to suggest that you can find the work done for any path.
Doesnt the path depend on the force field? So given F, there is a limited range of paths. (Depending on starting position, velocity, mass)
The integral is answering you the question “If the particle would move along this path, how large would be the amount of work done by it regarding this force field?”, while the path it actually will take depends on every force field there is and the initial values.
The set of paths on which you can use the integral does not tell you anything about the paths that make physically sense.