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MathGroup Archive 2006

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Re: Re: simplify a trig expression

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65529] Re: [mg65483] Re: simplify a trig expression
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Thu, 6 Apr 2006 06:52:43 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200603311109.GAA15029@smc.vnet.net> <200604011038.FAA07301@smc.vnet.net> <200604020900.FAA01612@smc.vnet.net> <11D40ADD-9EC9-4DCE-B685-1CA00605B9B2@mimuw.edu.pl> <e0r0c9$mt$1@smc.vnet.net> <200604051055.GAA21649@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

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, 
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.  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.

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
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305


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