Highlights • A new integration method for X-FEM elements cut by an interface is proposed. • The Heaviside function is replaced by an equivalent continuous higher order polynomial function. • A generic numerical strategy to compute this polynomial function is proposed. • This new integration method integrates correctly mechanical fields in elasticity and plasticity. Abstract The introduction of plasticity in an extended finite element (X-FEM) framework is very recent and arises several technical and scientific difficulties due to the moving position of integration points as the crack proceeds if a fixed grid is not used. Currently, the existing analyses in plasticity with X-FEM are limited to monotonic plasticity approximated by non-linear elasticity using the HRR field decomposition. However, such an analysis cannot be applied in fatigue where cyclic plasticity plays a major role in crack propagation. The aim of this study is to tackle one of the difficulties induced by cyclic plastic computation: the projection of the internal variables during crack propagation. The discontinuous Heaviside function in the elements cut by the crack is replaced by a new continuous enrichment function. This new function is polynomial and its degrees and coefficients depend on the type of element and on the position of the interface in the element. The automatic determination of the new enrichment and its introduction in a linear and non-linear framework have been implemented for linear elements in two dimensions and three dimensions and tested with convergence tests in elasticity and plasticity.