Dan350,
You need to know which force is doing the work (of in this case, which force work is being done against) to figure out what F is. Hooke's Law applies to "restoring forces" which pull objects towards an equilibrium position. The key point about Hooke's Law forces is that whether you are to the felt or right of the equilibrium point, or above or below it, the force points towards that point. Springs pull you back or push you forward, depending on which direction the equilibbrium point of the spring is, and as a result springs obey Hooke's Law.
Your problem has something pulling up a chain. So to figure out which force you are working against, you need to see which force is pulling in the other direction. You pull up and ______ pulls down. I think we can agree that _____ is gravity here.
So F = weight (measured in lb). However, you do not know weight. Instead, you are given (linear weight - or weight per unit length) which is weight/L, the weight divided by the length of the chain. F is not constant in your problem, because the L gets shorter as you continue to pull. The fact that the force is not constant tells you that you must use an integral (it is true that you must also integrate when you use Hookes Law, because F=kx, and so it is not constant but depends on x).
Mass = Linear density * Length, so you have F = (linear weight)*L. It is L that changes as the chain is pulled up, so you are integrating over dL. The smallest L (lower limit) is 0ft and the largest (upper limit) is 300ft.
\int_0^{300}\!\ (Linear Weight) * dL
Linear Weight is 2 lb/ft. There is no x inside the integral here.
