Choquet integrals are a class of non-additive aggregation operators that generalize the weighted arithmetic means and order statistics (for example, the sample median). It is a prevalent example of fuzzy integrals that has been extended in various ways. In particular, replacing the naive difference between assessments in the Choquet integral by restricted dissimilarity functions produces d-Choquet integrals. We define and study differentially private d-Choquet integrals. They are designed to satisfy differential privacy with the introduction of Lagrangian noise calibrated by their sensitivity. In this way, a new additive noise differential privacy mechanism is proposed that generalizes differentially private Choquet integrals. The later case already gave valuable information about the sensitivity of linear combination of order statistics and weighted means, for which the literature on privacy preservation is abundant. To meet the needs of practical implementation, we compute the sensitivity of d-Choquet integrals associated with several types of fuzzy measures and restricted dissimilarity functions. Other theoretical results link these achievements with knowledge about the particular case of differentially private Choquet integrals.