[CAS] Integrals

[AlexFekken]

(09-28-2017 11:15 AM)DrD Wrote:  For sure. However, the problem is not about "how" to solve the core issue. It is about the CAS integrate command, and how it handles f(x)=1/x^n, where x=0, over a continuous interval that includes 0, with even or odd powers.

I think you already partly answered that question in your original post. I simply suggested looking at those results in a wider context to try to understand what the tools does in those cases as well. At least I think the purpose of your question (which wasn't really a question, in spite of the question mark :-)) was to try to understand what the tool does with these types of problems.

As with most of these types of discussions, the expectation of many seems to be that we should be able to use a tool as a substitute for (rather than an aid to) our brain. And that we should not even have to state our questions unambiguously; the tool should just know what we want and infer any context (from what?). As in this case we typically discuss ambiguous situations in which there is a context or expectation that did not go into the question that we asked of the tool. And then we expect the tool to come up with the "right" answer neverthless.

In this particular case (integration), do we only want to see real (or complex) answers or do we accept extended real or complex numbers (and if so, using a one or two point compactification)? Do we want a Riemann integration (proper or improper) or a Lebesgue integration? I might want my integral of sin(x)/x from 0 to (plus) infinity to be undefined (or divergent) because I only care about "proper" Lebesgue integrals. Someone else might want to get the textbook result for the improper Riemann integral or "undefined" because it is not a proper Riemann integral. Similarly, I might want my 0^0 to be undefined because my exponents could be any (positive) real number, while someone else, working with power series perhaps, wants x^n to be 1 when n=0 and regardless of x.

Rather than expecting silver bullets, we should take responsibility for knowing the quirks and limitations (as opposed to outright bugs; the distinction may be difficult to make) of our tools, especially if we decide to blindly rely on them. If we can't agree on what outcomes we "should" get, then it is unreasonable to expect a tool to make the "right" decision for us.