Integral functionals of the form \[ I(u)=\int^{1}_{0}f(t,u')dt\quad u\in H^{1,q}(0,1;{\mathbb{R}}^ n) \] are considered. Here \(n\geq 1\), \(q\geq 1\), and \(f: (0,1)\times {\mathbb{R}}^ n\to {\mathbb{R}}\) is a nonnegative convex integrand. It is known [see the reviewer and \textit{G. Dal Maso}, Nonlinear Anal., Theory Methods Appl. 9, 515-532 (1985; Zbl 0527.49008)] that in this situation the \(L^ q\)-relaxed functional \[ \bar I(u)=\inf \{\liminf_{h\to +\infty}I(u_ h): u_ h\to u\quad in\quad L^ q(0,1;{\mathbb{R}}^ n)\} \] admits the integral representation \(\bar I(u)= \int^{1}_{0}g(t,u')dt\) for a suitable convex integrand g. Here the authors give an ``explicit'' way to construct the relaxed integrand g; more precisely they prove that \(g(t,z)=\sup \{p(t)z-f^*(t,p(t)): p\in K\}\) where \(f^*(t,\cdot)\) is the convex conjugate function of f(t,\(\cdot)\) and K is a suitable countable subset of \(H^{1,q'}\) \((1/q+1/q'=1)\). In Section 3 the procedure above is applied to compute the relaxed integrand g in many critical examples.