If you have trouble understanding this, then the best thing you can do is to replacethe problem by simpler problems and see if you can solve them.
E.g. try to integrate the function:
Assuming that you are given these problems for homework, the purpose of solving the problems is not to get the right answer, it is for you to learn to solve integrals. So, it doesn't really matter what problems you solve to learn it. But trying to solve a problem for too long is an enormous waste of time.
I'm not quite sure about your explanation on Partial Fractions... And Liouville's Therom. We haven't done that in class... so...
Well, I think you just need to look at a simpler case. The algebra may sometimes obscue things. Also, you don't need to understand Lioville's theorem. Also, you don't need to stick to what you've been taught in class.
All you need to do is pick up a piece of paper and try to solve some problems. E.g. try to expand the following functions in partial fractions (using any method you like):
1) A/x + B/(1+x)
2) A/x + B/x^2 + C/(1+x)
I think I'm right....
You now just need to solve for A, B, and C :)
That's my problem. When I tried solving for A, B, C, D, and E in that first problem, it didn't really work out right...