The goal of this paper is to find out under which conditions the resolvent R(λ,A) of the generator A of a positive semigroup T = (T(t))_t≥0 on the space L^P(Ω) (1≤p ≤∞) is an integral operator. For this purpose we investigate the integral representability of some integrals of operator-valued functions (Theorems 2.1 and 2.10).