Defining a new variable $u=t-x$, we can rewrite the principal value integral as $$\begin{equation} I_{\Delta}(x)={\cal P}\int_{-\Delta}^{+\Delta}du\frac{f(u+x)}{u}. \tag{19} \end{equation}$$ One possibility is to Taylor expand $f(u+x)$ around $u=0$, and compute derivatives to a certain order as we did for the Trapezoidal rule or Simpson's rule. Since all terms with even powers of $u$ in the Taylor expansion dissapear, we have that $$\begin{equation*} I_{\Delta}(x)\approx \sum_{n=0}^{N_{max}}f^{(2n+1)}(x) \frac{\Delta^{2n+1}}{(2n+1)(2n+1)!}. \end{equation*}$$