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Re: simplify a trig expression

Murray Eisenberg <murray at math.umass.edu> wrote:
> I don't think the the example of Integrate[1/x, x] is strong evidence
> that the effot is "doomed to failure".
>
> The usual answer Log[Abs[x]] that most textbook teach is rather silly,

OK, that's your opinion. It doesn't happen to be mine.

> since it suggests (although of course does not logically imply) that one
> could use that formula together with the Fundamental Theorem of Calculus
> to evaluate Integral[1/x, {x, a, b}] over an interval with a < 0 < b.
> And of course the result would be nonsense because neither Integral[1/x,
> {x, a, 0}] nor Integral[1/x, {x, 0, b}] converges.

True, they don't converge. But it's interesting that, if one does what you
say is suggested, then the result does make perfectly good sense as a
Cauchy principal value. [Note that I'm _not_ saying that the method is
correct, that you should mention Cauchy principal values to your class,
etc.]

> I would much prefer
> if textbooks would say, like Mathematica, that Integrate[1/x] has value
> Log[x] over intervals of positive reals, whereas Integrate[1/x] has
> Log[-x] over intervals of negative reals; the absolute value function in
> this context, in my experience, just obfuscates the issue.

Can we get Mathematica to say that Integrate[1/x, x] is something like

Which[x > 0, Log[x], x < 0, Log[-x]]  ?

It seems to me that's what you'd like when x is real, but surely we can't
get Mathematica to give us such.

Look below at what you said you were trying to do. I still say that it
must be doomed to failure. But I can't see that you need to do that. My
main point in my previous response was in my last paragraph: If you can
depend on the mathematical intelligence of your graders, then you don't
need to have Mathematica give answers in a particular form.

Regards,
David Cantrell


> David W. Cantrell wrote:
> > Murray Eisenberg <murray at math.umass.edu> wrote:
> >> Actually, what I was trying to do is this:  To obtain in Mathematica,
> >> the answers to a ten-question integration exam that would be of the
> >> form students would obtain with standard paper-and-pencil techniques.
> >> And the purpose of that was to to provide to the graders, whom I
> >> supervise, answers that are unquestionably correct -- and, again, in
> >> that form.
> >
> > But surely this endeavor is doomed to failure. Consider the simple
> > example, which happens to be closely related to your earlier ones:
> >
> > Integrate[1/x, x]
> >
> > Mathematica will give just Log[x], which is perfectly correct. But I
> > presume that you want your students to give Log[Abs[x]] plus an
> > arbitrary constant of integration. How might one get Mathematica to
> > give Log[Abs[x]]? I certainly don't know how.
> >
> > It seems to me that you should give your graders _one form_ of correct
> > answer for each problem and that you must then depend on their
> > mathematical intelligence to recognize alternative correct forms. If
> > you can't depend on that, are they really qualified to be graders?
> >
> > Regards,
> > David Cantrell
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