We present numerical methods for the approximation of singular and nearly singular integrals arising from 2D problems in classical potential theory. We begin with third-order approximations of integrals involving logarithmic singularities. We present a new proof, using the Euler-Maclaurin summation formula, for the accuracy of a formula for approximating integrals with logarithmic singularities on the real line. It consists of a Riemann sum excluding the point of singularity plus a special correction term. We extend this formula to single layer potentials at points on a smooth closed boundary where singularities arise in the Green's function in the integrand. Next we use a technique of smoothing the Green's function to compute the single layer potential at points on or near a smooth closed boundary, where the Green's function will become singular or nearly singular. We derive correction terms to add to a trapezoidal sum for the potential to obtain third-order accuracy.

We develop a second-order method for approximating double layer potentials on a curve segment or piecewise smooth closed curve with corners, evaluated at points on or near the curve. The potential kernel is smooth for points of evaluation on the curve, but is nearly singular for points near the curve; special care is needed near the endpoints. The method uses trapezoidal sums with corrections from the Euler-Maclaurin summation formula. For points near an endpoint of the curve segment, we present a new approximation method in which we split the segment into a part near the point of evaluation and a part away from the point of evaluation. We approximate the integral separately on each part. For the part near the point of evaluation, we use a linearized approximation to derive a correction in the form of an exact integral minus a sum. The remaining part is sufficiently far away from the point of evaluation so that the trapezoidal sum with corrections from the Euler-Maclaurin formula is accurate. For points close to the curve but away from the endpoints, a previously derived correction formula for smooth closed curves is used. We present estimates to verify that in all cases a second-order accurate approximation is obtained.