The numerical integration of expressions containing strong singularities or strong near-singularities has long been a challenging problem in the electromagnetics community. Much attention has been paid to this problem, as strong 𝟏/𝑹 𝟐 singularities routinely appear when implementing electromagnetic simulation techniques like the Method of Moments (MoM). To date, several techniques, from singularity extraction to singularity cancellation, have been employed to deal with problems of this type. However, no single technique has been proposed that can deal with both strong singularities and strong near-singularities in a fully-numerical manner. Moreover, it has been claimed that the Helmholtz-type strongly singular integral found in the MoM is convergent in a principal value sense, but this has yet to be proven rigorously. In this work, we will conduct the convergence proof and introduce a "polar scaling" change of variables method that may be used to evaluate Helmholtz integrals with both strong and weak singularities/near-singularities. The technique is fully-numerical and can in principle be applied to any planar polygon and any basis function. We will also provide numerical results showing useful convergence behavior for integrals involving both exact and near-singularities.