Let (Ω,μ) be a measure space, E be an arbitrary separable Banach space, $E{*}_{\omega *}$ be the dual equipped with the weak* topology, and g:Ω × E → ℝ be a Carathéodory function which is Lipschitz continuous on each ball of E for almost all s ∈ Ω. Put $G\left(x\right):={\int }_{\Omega }g(s,x\left(s\right))d\mu \left(s\right)$. Consider the integral functional G defined on some non-$L^p$-type Banach space X of measurable functions x: Ω → E. We present several general theorems on sufficient conditions under which any element γ ∈ X* of Clarke’s generalized gradient (multivalued C-subgradient)...