Will Lecture Videos on YouTube Appeal to a Popular Audience?

Thanks – I asked my Twitter followers for some Mathematica-related results but they all came back hypergeometric. Maybe they weren’t all trying a forced simplification. 

Amazingly, Mathcad gives the correct answer for the e^(-t^3) integral.

From Maple 15 command line interface:

> diff( int(-t^3,t=0..x), x );  
                                        3
                                      -x

The FTC seems to work OK here.  When we separate the integration and differentiation operations is where we get into trouble.

There are actually a lot of questions and issues raised here.
Why didn’t Maple just apply the FTC?
The evaluation rules for Maple mean that Maple is computing an antiderivative before it
differentiates. To avoid this, try the command Int instead of int.
Int does not evaluate so Maple is not trying to anti-differentiate. When you try to
differentiate something like
Int(exp(-t^3),t=0..x) then Maple applies the FTC correctly. Try it!
Why does Maple use hypergeometric functions? One reason is because not all functions have antiderivatives in elementary terms – involving just exp, log, trig and rational polynomials. So you have to use special functions and as my friend Tony Scott who developed many of these techniques described it to me – they allow one to find antiderivatives for almsot all the functions that typically show up in physics – even high level physics. Why can’t one easily simplify the result with hypergeometric functions to the answer expected from the FTC? Simplification is a really hard problem that has no precise solution.How do you define what is simple? A related hard problem is recognizing equivalent expressions. There are no known good solutions for these problems. Maybe your calculus students need a thesis problem for their Ph.D. in computer science? Why doesn’t Maple change it’s evaluation rules to use the FTC when possible?  Maple (like most other math software) is a multi-purpose tool that can’t
possibly anticipate the circumstances from which you ask this question.
Some users actually wanted that hypergeometric function as an answer.Couldn’t there be a Maple for Calculus that avoids higher functions and just responds in a way that is appropriate for a calculus student? Many people have tried to build math software (or alter existing software) to do just that. Maple
has tools to do that, too. Other software have done this and Derive is exceptionally good at it. But fundamentally, it is an IMPOSSIBLE task!

If you know algebra, you know that you can’t work with polynomials over the
integers without complex numbers lurking somewhere in the background. Ask
the right Calculus style question and it becomes impossible to answer it
without introducing complex numbers and maybe hypergeometric functions and
a hell of a lot more math, too.

Many people who know about this use it as a teaching opportunity. It should whet the student’s
appetite to learn more.

Try the following in WolframAlpha

derivative with respect to x of integral of exp(sin(t)) from t=0 to t=x

Now try this in Maple

diff(int(exp(sin(t)), t = 0 .. x), x)

Suddenly Maple knows the FTC while WolframAlpha has forgotten about it.
I think Blair’s comment is up to the point. Symbolic computation is extremely complicated, and finding examples that do not work as expected in a given software is not uncommon, but in general Maple, Sage, Mathematica and WolframAlpha are excelent tools.

One thing I do not like in WolframAlpha is that you cannot restrict your functions to the reals easily (try plotting log(x) and explain to your calculus students the result)  

It’s nice to see a Chronicle blog on something other than students’ writing deficiencies!

You state that computer algebra systems can only do math computationally, and not conceptually. That is not a very good characterization. Maple will successfully inform us that the derivative w.r.t. `x` of the inert integral Int(exp(-t^3),t=0..x) is exp(-x^3) and this is more conceptual Calculus than it is computational. Note the use of the capital “I” in `Int`, for the inert form. Indeed, for an unknown integrand g(t) Maple can also succeed here. But you did not ask for the inert integral. Rather, you have used the active form (with lowercase `int`) which does a computation of the integral before passing off to the `diff` command. In other words, it is you the user who has forced the more “computational” behaviour by Maple. It is partly on account of such distinctions that Maple uses active forms like `int`, `diff`, `sum`, and `product` while also having inert forms such as `Int`, `Diff`, `Sum`, etc.
You stated that the correct result for the second problem was exp(-t^3), but actually the correct result is exp(-x^3).
On the topic of conceptual aspects of calculus, the reverse question is also interesting. For an unknown `g` Maple will return unevaluated for the command int(diff(g(t),t),t=0..x), but will return g(x)-g(0) for the more specific command int(diff(g(t),t),t=0..x, continuous).
Maple’s knowledge of abstract Calculus is certainly not perfect! Analysis and Calculus are very hard things to teach a CAS to do well.
ps. Here is a link to a webinar on introductory symbolic integration in Maple. (The presenter starts slowly, but it becomes more confident and interesting. It begins with indefinite integration, and then switches over about halfway to discuss definite integration.) http://www.mapleprimes.com/posts/97477-Webinar-On-Symbolic-Integration