One dimension conservative force and potential energy

[SPhy]

Homework Statement

Given a conservative force, how can we obtain the change in potential energy?
Given a potential energy function, how can we determine the associated conservative force?

One dimensional.

Homework Equations

F_x = -du/dx

ΔU = -∫ F dx

The Attempt at a Solution

I know I can use the above expressions, however, how can I arrive at these expressions? My textbook and class notes simply say

BY DEFINITION: We will say the change in potential energy is equal to negative of the work done by the internal conservative force.

I know ΔKE = ∫ F ⋅ dr

I can integrate the right side but I obtain something like G(x₁) - G(x₂)

But then we define U(x) = -G(x) ?

So then to answer the question, should I simply use my equations listed above? Or is there a better way to approach this?

 

[haruspex]

I know ΔKE = ∫ F ⋅ dr

There is not necessarily any KE involved here. That's just one possibility for where the work done has gone. If you generalise this to work done = ∫ F⋅dr then you seem to have all you need.