= Solution
For singularities $a<c_1<\cdots<c_N<b$, the Cauchy principal value is
$$
\boxed{
\operatorname{PV}\int_{-\infty}^{\infty}f(x)\,dx
=\lim_{\varepsilon\downarrow0}
\left[
\int_{-\infty}^{c_1-\varepsilon}f(x)\,dx+
\sum_{j=1}^{N-1}\int_{c_j+\varepsilon}^{c_{j+1}-\varepsilon}f(x)\,dx+
\int_{c_N+\varepsilon}^{\infty}f(x)\,dx
\right]},
$$
where the assumed tail <integrals> make the two unbounded pieces meaningful. Equivalently, remove a symmetric interval of radius $\varepsilon$ about each $c_j$ and take the <limit> of the remaining <integral>.
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