In this paper, we compute the local integrals, with normalized unramified data, over a p-adic field F, arising from general Rankin–Selberg integrals for SOm × GLr+k+1, where the orthogonal group is split over F, k≤[m−12], and the irreducible representation of SOm(F) has a Bessel model with respect to an irreducible representation of the split orthogonal group SOm−2k−1(F). Our proof is by “analytic continuation from the unramified computation in the generic case”. We let the unramified parameters of the representations involved vary, and express the local integrals in terms of the Whittaker models of the representations, which exist at points in general position. Then we apply analytic continuation and the known unramified computation in the generic case. We discuss some applications to poles of partial L-functions and functorial lifting.