Summary: Relaxation theorems for multiple integrals on \(W^{1,p}(\Omega;\mathbb R^N)\), where \(p\in ]1,\infty[\), are proved under general conditions on the integrand \(L:\mathbb M\to [0,\infty ]\) which is Borel measurable and not necessarily finite. We involve a localization principle that we previously used to prove a general lower semicontinuity result. We apply these general results to the relaxation of nonconvex integrals with exponential-growth.