I am learning about cauchy principal value integrals and was looking for help evaluating the integral given by
\begin{align} \lim_{\epsilon \to 0} \int_{-a}^{-\epsilon} \dfrac{A}{x} dx + \lim_{\epsilon \to 0}\int_{\epsilon}^a \dfrac{B}{x} dx \end{align}
where $A$ and $B$ are constants. My attempt so far is:
\begin{align} \lim_{\epsilon \to 0} \int_{-a}^{-\epsilon} \dfrac{A}{x} dx + \lim_{\epsilon \to 0}\int_{\epsilon}^a \dfrac{B}{x} dx = \lim_{\epsilon \to 0} [A(\ln \epsilon - \ln a)+B(\ln a - \ln \epsilon )] \end{align}
If $A = B$, then I can cancel out the $\ln \epsilon$ terms and the principal value is 0. However, I would like to know how to determine the principal value if $A \neq B$.
If $A > B$ the the principal value is $-\infty$. If $A < B$ then the principal value is $+\infty$.