In this link, page 252, there is an equality of interest, (D.69)
$$\int_{-\infty}^{\infty}\dfrac{x\cdot J_0(x\cdot r)}{x^2-k^2}dx=\left\{ \begin{array}{cc} \dfrac{i \cdot\pi \cdot H_0^{(1)}(k\cdot r)}{2}&\text{Im}(k)>0 \\ \\ \dfrac{i \cdot\pi \cdot H_0^{(2)}(k\cdot r)}{2} &\text{Im}(k)<0\\ \end{array} \right. \tag{D.69}$$
Where $J_0$ is the BesselJ, $H_0$'s are the Hankel functions respectively.
How to prove the integral equality?
And how to obtain a numerical solution of such integrals? e.g., I tried to let $k=10$ and $r=6$, but found most of the available integral algorithms didnot lead to convergence probably due to the existence of a discontinuity point $x=k$ of the integrand and the improper upper limit.