Why Are There Three Different Integrals for Hooke's Law?

[eseefreak]

I'm trying to understand the math behind Hooke's law and work done by a spring. I'm really looking for clarity. I am trying to understand why after integrating the equation I get three different results. I understand that each formula is specific to a problem but why does this work mathematically? I attached an image of the three formulas together.

Thank you. Much appreciated

 

[nasu]

I think they are a little confusing (or confused).
What they call "specific" is actually the general case, the work done by the elastic force between any two arbitrary positions. The first expression is just what you get from the general one, for the particular case of initial position xf=0 and xi=x_max.