Although improper integrals constitute a concept of great utility for Mathematics students, it appears that students are unable to assimilate this concept within the wider system of concepts they learn in their first year of Mathematics studies. In this paper we describe a competence model used in a study about the kind of understanding students possess about improper integral calculus when two registers of representation come into play. Competence will be considered as the coherent articulation of different semiotic registers. After analysing the results of a questionnaire, six students were selected to be interviewed on the basis of their overall results and the significance of their answers. For the interview, five original questions from the questionnaire were used together with a new question. In this article we will analyse, from our theoretical point of view, the work carried out by one student who was interviewed to show how our competence model works and we will discuss this formal competence model used.