Recently, it has been noted that some regular one-dimensional integrals, whose minimizers in an appropriate class of absolutely continuous functions, have unbounded derivatives at certain points. These singularities may prevent the minimizers satisfying the classical conditions of calculus of variations. To compute these singular minimizers in one-dimensional problems, a numerical method is suggested and discussed. The scheme proposed is shown to converge to an absolute minimizer and is tested on an example. The effect of quadrature is analyzed. Implications for higher-dimensional problems in nonlinear elasticity are discussed.