Much attention has been given to a possible violation of the optical sum rule in the cuprates and the connection this might have to kinetic energy lowering. The optical integral is composed of a cutoff-independent term (whose temperature dependence is a measure of the sum-rule violation), plus a cutoff-dependent term that accounts for the extension of the Drude peak beyond the upper bound of the integral. We find that the temperature dependence of the optical integral in the normal state of the cuprates can be accounted for solely by the latter term, implying that the dominant contribution to the observed sum-rule violation in the normal state is due to the finite cutoff. This cutoff-dependent term is well modeled by a theory of electrons interacting with a broad spectrum of bosons.