Abstract A novel method is developed for the solution of integral equations.An analytic solution is obtained by exponential approximation of the kernel; and to this solution is added the solution of an auxiliary integral equation. Since the auxiliary equation deals with magnitudes that are only a few per cent of the original quantities, a crude numerical solution of the auxiliary equation is satisfactory. In fact, sufficient accuracy is obtained in many engineering applications by ignoring the auxiliary equation completely: all that is then needed is an indication of the maximum error caused by approximating the kernel. Simple equations are derived for the magnitude of this error. The results are applied to interflections in cylindrical blackbodies, and the errors are evaluated for previous solutions of the problem by Buckley and by Yamauti.