I've been studying the method of partial fractions for evaluation integrals. So far every example and exercise have been fairly straight forward, but I still have some unanswered questions:
1) The standard table for doing method of partial fractions - I guess some clever person has come up with this table and verified it works. However is this the only way of doing partial fractions, or are there other tables / methods around ?
2) The system of equations to be solved from using method of partial fractions - Does this system always has a solution ? So far I've never experienced a system with no solutions, but that is probably because classroom exercises are solvable. Can one come up with a system of equations not solvable for constants $A$, $B$, ... $K$, such that method of partial fractions fails ?
Thanks for your time.