© 2017 Elsevier Inc. Let X and Y be Banach spaces and let ? be a compact Hausdorff space. By the classical Bartle–Dunford–Schwartz theorem, any operator S?L(C(?),Y) admits an integral representation with respect to a Y??-valued measure. By the Dinculeanu–Singer theorem, each operator U?L(C(?,X),Y) admits an integral representation with respect to an L(X,Y??)-valued measure. We establish an integral representation for any operator S?L(C(?),L(X,Y)) with respect to an L(X,Y??)-valued measure. This far-reaching extension of the Bartle–Dunford–Schwartz theorem serves as a departure point for a general integral representation theory developed in the present paper. In particular, it is an efficient tool that enables us to give an alternative simple proof to the Dinculeanu–Singer theorem. The latter theorem is proved in a more general context of X-valued continuous functions with p-compact range. Among others, useful formulas which connect different vector measures are deduced.