We define the notion of weak universal integral based on a semicopula, and introduce and discuss two particular classes of weak universal integrals based on semicopulas, which generalize the well-known Sugeno and Shilkret integrals. In special cases, when the considered semicopulas are bounded from above by the Łukasiewicz t-norm, all introduced integrals reduce to the corresponding smallest semicopula based universal integrals. Remarkably, when the product semicopula is considered, the proposed integrals generalizing the Shilkret integral belong to the class of aggregation functions, which is not the case of the minimum semicopula when the introduced integrals generalize the Sugeno integral.