Local Lipschitz continuity of minimizers of certain integrals of theCalculus of Variations is obtained when the integrands are convex withrespect to the gradient variable, but are not necessarily uniformlyconvex. In turn, these regularity results entail existence of minimizers ofvariational problems with non-homogeneous integrands nonconvex withrespect to the gradient variable. The x-dependence, explicitly appearingin the integrands, adds significant technical difficulties in the proof.