Given a sequence of integrands f(n) : T x X --> R (n greater than or equal to 1) which converges in the sense of the slice-topology to an integrand f, ( T, A, mu) being a complete probability space and X a nonreflexive Banach space with separable dual, we show that the sequence of integral functionals I (f (n)) : u --> integral f(n) (t, u (t))d mu (n greater than or equal to 1) associated to the (f(n)) converges to I (f) : u --> integral f(t, u (t))d mu in the sense of the slice-topology on L-p ( X) and that the sequence of integral functionals associated to the conjugate integrands (f*(n)) converges to I (f*) : u --> integral f*(t, u ( t))d mu on L-q (X*) (with 1 less than or equal to p < +infinity and p(-1) + q(-1) = 1). This is an extension of some results which were shown to hold by Joly and de Thelin for Painleve-Kuratowski convergence when X is finite dimensional and by Salvadori for Mosco convergence when X is reflexive. We also need to provide some criteria for functional convergence in the slice-topology, using the strong epigraphical upper limit.