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We study a 1-parameter family of trigonometric definite integrals, showing how the joint usage of Information and Communication Technologies and paper-and-pencil work lead to different outputs, revealing different mathematical meanings and different concrete meanings. This family of integrals is useful for describing a phenomenon in soil mechanics, whence the importance of such integrals in STEM education.

We present a sequence of improper integrals, for which a closed formula can be computed using Wallis formula and a non-straightforward recurrence formula. This yields a new integral presentation for Catalan numbers.

We compute closed forms for some parametric integrals. The tools used are MacLaurin developments and other infinite series. Finally, Stirling numbers appear. (Contains 1 figure.)

Looking for closed forms for an integral depending on one parameter, we obtain integral identities and new integral presentations of Catalan numbers. Wallis formula appears as an important tool.

We compute in three different ways the same definite parametric integral. By-products are the derivation of a combinatorial identity and two integral presentations of Catalan numbers. One of them leads to a presentation using the [gamma] function.

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.

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

When a professional engineer solves a mathematical problem using technology, he/she generally uses a single one-step "high-level" command of a readily available computer package to obtain the solution immediately. This should not happen during his/her learning cursus. In an engineering mathematics course, the educator should decompose the solution process into elementary steps and reinforce each step by a "low-level" usage of a CAS. This approach is absolutely essential in order to provide the future engineer with the conceptual insight into the solution process. We illustrate this with the teaching of the solution of ordinary differential equations using Laplace transforms.

Closed forms are computed for parametric integrals, generally using induction formulas. It is shown that these integrals can be core activities, mixing hand-work, computations with a computer algebra system and experimental mathematics with an interactive website.