newtonian mechanics - Is work done = change in KE, or change in mechanical energy?

Starting from $F = ma$ we get:

$$F=m\frac{dv}{dt} = m\frac{ds}{dt}\frac{dv}{ds} = mv\frac{dv}{ds}$$

which leads to the work done:

$$W = \int Fds = \int mvdv = m\frac{v^2}{2} - m\frac{u^2}{2} \tag{1}$$

which is equal to the change in KE.

If a body is lifted against gravity by a variable force which is zero at time $t=0$, and also zero after having lifted the body a distance $h$, then is it correct that the work done against gravity is $mgh$?

Why does this not violate the equation (1) above? Why does the PE does not appear in equation (1)?