We describe an efficient method for extracting the parts of D-dimensional loop integrals that are needed to derive observables in classical general relativity from scattering amplitudes. Our approach simplifies the soft-region method of integration by judiciously combining terms before the final integrations. We demonstrate the method by computing the required integrals for black-hole scattering to the second post-Minkowskian order in Einstein gravity coupled to scalars. We also confirm recent results at the third post-Minkowskian order regarding universality and high-energy behavior of gravitational interactions in maximal supergravity.