New $L^p$-Integral Inequalities Involving the Primitive of a Function

We present two new general \(L^p\)-integral inequalities involving the primitive of a function. They extend the applicability of some established results in the literature, mainly thanks to the presence of an intermediate function that can be adapted to the context. We thus contribute to the understanding of the interaction between the \(L^p\)-integrals of functions and their primitives.