I know how to do these problems, but this one is giving me some trouble:
Expand the following into partial fractions$$\frac{5}{(x+1)(x^2-1)}$$
Should I break it down into: $$\frac A{x+1}+\frac B{x-1}+\frac C{x+1}\tag{1}\label{1}$$or$$\frac A{x+1}+\frac B{(x+1)^2}+\frac C{x-1}\tag{2}\label{2}$$
I tried it both ways and came up with different possibilities for the constraints:
If it is broken down like \eqref{1}, then
$$\begin{cases}A+B+C=0\\
2B = 0\\
A+B-C=5 .\end{cases}$$
If it is broken down like \eqref{2}, then $$\begin{cases}A+C=0\\ B+2C=0\\ A+B-C=5 .\end{cases}$$