The goal in this paper is to prove an existence theorem for the solutions of a class of functional integral equations which contain a number of classical nonlinear integral equations as special cases. Our investigations will be carried out in the space of continuous and bounded functions on an unbounded interval. The main tools are some techniques in analysis and the Schauder fixed point theorem via measures of noncompactness. Our results extend and improve some known results in the recent literature. Three nontrivial examples explain the generalizations and applications of our results.