Some integrals with problematic evaluation

[quinyu]

Once again, we agree that many math teachers simply don't take the time to understand what's going on under the surface. Nevertheless, for the general high school teacher, it might well be a bit too high expectation to know and fully understand for example the Hermite-Kronecker-Brioschi characterization (which in itself isn't a particularly complicated one, at least in terms of programming), and know the associated caveats when it may or may not be used, or even the path to reduce a quintic equation to a form suitable for the method. From that point on, conveying the same bulk of information to the high school students might border on applied sadism. And strictly speaking, the above-mentioned method is still in the field of algebra, not that of calculus; and not something that'd make its way into the Journal of Symbolic Computation.

But I believe I understand your point of view as well; limitations, particularly ones without a good reason, can often be very annoying, calculators or otherwise. Nevertheless, maybe there's a middle way, allowing the user himself to limit what the calculator is doing for him. In the HP, or any other handheld, the methods are hidden behind smoke and mirrors (and for this reason, I particularly praise this recent development in xcas.) Expose that interface - let it be configurable what the calculator itself is willing to do or not. For the students who have insight, this can be a very practical way of taking things at their own pace, or teachers who want to influence more precisely what may happen in exam mode - right now it is rather coarse-grained. I already know (or expect the student to know) how to solve a standard quadratic equation - let the calculator do that for me, it's just boring number crunching from that point on. I didn't yet learn (or teach) how to calculate derivatives of an expression containing hyperbolic functions - turn it off.

The whole point of computers in general was to take the task of boring number crunching off the shoulder of humans who can instead focus on the parts of the task requiring intuition or insight (which computers lack.) The math teacher likely expects no rote learning from the students (excepting perhaps such basic things as a multiplication table or some basic identities), but the development of the skill to see where a method or another might be useful, regardless of the field. In a manner, the handful of "tutors" that are included in Maple are a good step in that direction. The tutor isn't going to tell you what to do with the integrand (unless you ask); you can apply any of the methods listed, if you so choose, and if it's at all applicable. In a while, the student learns to see which option is viable in any given situation.

And while Maple certainly can do integration by simply using the command "int," in my point of view, the real satisfaction comes from gaining the insight and applying it properly; and perhaps you can ultimately arrive to a neater form than that produced automatically by the CAS. Perhaps not; but you can compare and learn from it, particularly if you see the stepwise approach, assuming it's human-compatible. I wouldn't call it human-compatible, for example, to spot that a part of your infinite series is actually rewritable as the sum of three Taylor series and simplify it accordingly. Maybe, when you've been in that particular field for several years, it's doable, but for everyone else... it's not their game.