How do you integrate the function $\frac 1{x^2 - a^2}$, where a is a constant? My problem with integrating by parts is that it always ends up being recursive, as I feel like I'm going in loops. Would appreciate it if someone can help me understand the process. 
$$\int \dfrac 1{x^2-a^2}dx$$
I know the answer is supposed to be $\dfrac 1{2a}\ln\left|\dfrac {x-a}{x+a}\right|+C$, but I can't figure out how to derive it.

You have to compute $A$ and $B$ such that
$$\frac1{x^2-a^2}=\frac A{x-a}+\frac B{x+a}$$
To do that, write
$$\frac A{x-a}+\frac B{x+a}=\frac{Ax+Aa+Bx-Ba}{x^2-a^2}$$
Then we have an equality of polynomials: $Ax+Bx+Aa-Ba=1$
We conclude that:
$$A+B=0$$
$$Aa-Ba=1$$
Solve this system for $A$ and $B$.
