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We compute closed forms for two multiparameter families of definite integrals, thus obtaining combinatorial formulas. As a consequence, a surprising formula is derived between a definite integral and an improper integral for the same parametric function.

We describe and analyze a situation involving symbolic representation and graphical visualization of the solution of a system of two linear differential equations, using a computer algebra system. Symbolic solution and graphical representation complement each other. Graphical representation helps to understand the behavior of the symbolic solution. Together with it, the symbolic solution and its analysis are needed to understand the graphical representation and to overcome the limiting constraints of the CAS. The study described in this paper points out the importance of flexibility in building connections between different mathematical registers, together with the fact that the balance between the usage of symbolic representations and of graphical representations can be very different from one student to the other.

The study of a real function of two real variables can be supported by visualisation using a Computer Algebra System (CAS). One type of constraint of the system is due to the algorithms implemented, yielding continuous approximations of the given function by interpolation. This often masks discontinuities of the given function and can provide very strange plots. The pixelisation of the visualisation adds to this strangeness. We present a study of such a "distortion of reality", using more than one system, and show how they can be used as a starting point to learn more Mathematics. This can be an obstacle, but we see it as a chance, as it is a motivating constraint, both for students and for educators. (Contains 10 figures.)

The constraints of a computer algebra system (CAS) generally induce limitations on its usage. Via the pedagogical features implemented in such a system, "motivating constraints" can appear, encouraging advanced theoretical learning, providing a broader mathematical knowledge and more profound mathematical understanding. We discuss this issue, together with two examples from Calculus, which show an important feature of an instrumentation process.

Despite the ready availability of computer algebra packages, from a pedagogical point of view, the authors still feel that integration should be taught extensively the classical way, by means of carefully selected examples which combine as many fundamental techniques as possible for their evaluation. Furthermore, whenever possible, these examples should be specifically selected from those which a Computer Algebra System (CAS) either cannot solve precisely to provide an exact answer in analytical or transcendental form, or which a CAS cannot solve altogether. This approach ensures that students gain a proper understanding of the topic and, perhaps even more important, the fact that the students can evaluate precisely in closed form integrals which the computer cannot, stimulates and motivates them further to study these techniques. This note illustrates the above claims by means of both an indefinite integral and the associated improper integral which is offered to students as the 'integral of the year'.

For questions in calculus, tables containing either intermediate or final results provide the student with a semi-graphical guide both to enhance their intuition and to help complete tasks. The joint usage of (hand-made) tabular presentations and of a CAS, in particular its graphical features, is particularly valuable in step-by-step working: such a learning process provides a more profound understanding of the mathematics involved and helps to develop better skills for solving problems. The examples considered include plane curves, either graphs of functions or parametric curves and space curves in implicit, parametric or polar form. [ABSTRACT FROM AUTHOR]

In a previous paper, we showed the importance of using low-level commands of a CAS in order to enhance the students' conceptual insight into mathematics. In this paper, we wish to emphasise a different use of a CAS, which makes possible the exploration of new topics, when the necessary theoretical results have still not been acquired, and avoids the blockage due to this lack of theoretical background. We illustrate this with examples from calculus. [ABSTRACT FROM AUTHOR]

In a previous note \[2], some geometric constructions were shown which can be interpreted as a ray of light trapped in a polygon by successive reflections on the sides. The same question is addressed here, replacing polygons by conics or pairs of conics. In some cases, the light is concentrated towards a specific segment. The tools used for the computations are elementary geometry, calculus and, in some cases, Commutative Algebra Software. [ABSTRACT FROM AUTHOR]