In information fusion, aggregations with various backgrounds require a variety of integrals to handle. These integrals are generally nonlinear since the set functions used are nonadditive in many real problems. In this study, the set functions considered are nonnegative and vanishing at the empty set. They are a class of set functions including fuzzy measures and even imprecise probabilities. A new type of nonlinear integrals with respect to such a set function for nonnegative functions is introduced and its primary properties are detailed. These type of integrals has a natural explanation and, therefore, has wide applicability. We also show a comparison between the newly introduced nonlinear integral and other nonlinear integrals, such as the Choquet integral, the natural extension in the theory of imprecise probabilities, and the common pan-integral. With a flowchart, the algorithm for calculating the integral is given in this paper when the universe of discourse (the set of all information sources) is finite.