A new representation formula for the solution of Fredholm integral equations of the second kind. The formula has certain computational advantages over the usual representation involving the resolvent or the representation formula of Krein. It makes use of the phi function and requires two integrations. The usual method of solution uses either the Fredholm resolvent (three variables) or the Krein representation, which requires differentiation as well as integration. The Bellman-Krein partial differential equation for the resolvent can also be obtained using this new formula. 9 pp. Refs.