For a constant force, the force F can be taken out of the integral, to get the result

For our course we just define work in the simplest possible way: W_F = F_x Δx.

We can write W_F = ∫ F · v dt which is useful in making clear that the work done by any force is ZERO if either v = 0, or F · v = 0.

For our course just remember that the work done by any force is zero unless the force has a component along the direction of motion of the object.

No work is done either by the force n or the force mg.

For the fundamental forces of nature, such as gravity, the work done by or against the force is independent of the path taken.