In this article, we explore the properties of the two generalized Sugeno integrals that we obtain by substituting in the expression of a classical Sugeno integral, the Kleene-Dienes conjunction and the Kleene-Dienes implication by the Gödel conjunction and the Gödel implication, respectively.A major difference compared to the classical Sugeno integrals is that the implication-based Gödel integral and the conjunction-based Gödel integral do not return the same result. In this paper, we investigate the adaptation of classical results for Sugeno integrals to Gödel integrals. Namely, their elicitation according to a piece of data and the extraction of selection and elimination if-then rules. The selection rules are obtained from the focal sets of the capacity underlying a conjunction-based Gödel integral, while the elimination rules are extracted from the focal sets of the conjugate of the capacity defining an implication-based Gödel integral.To illustrate our results, we apply our constructions to a real data example already used for Sugeno integrals.