We consider the problem of minimizing autonomous, simple integrals such as (P) min{integral(0)(T) f(x(t), x'(t)) dt: x is an element of AC([0,T]), x(0)=x(0), x(T)=x(T)}, where f: R x R-->[0,infinity] is a possibly nonconvex function with either superlinear or slow growth at infinity. Assuming that the relaxed problem (P**) obtained from (P) by replacing f with its convex envelop f** with respect to the derivative variable x'-admits a solution, we prove attainment for (P) under mild regularity and growth assumptions on f and f**. W discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem (P**), and we present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp.