Re: simplify a trig expression

*To*: mathgroup at smc.vnet.net*Subject*: [mg65594] Re: simplify a trig expression*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Mon, 10 Apr 2006 02:31:14 -0400 (EDT)*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$1@smc.vnet.net> <200604051055.GAA21649@smc.vnet.net> <e12tb8$jfc$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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

**References**:**Re: simplify a trig expression***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: simplify a trig expression***From:*Murray Eisenberg <murray@math.umass.edu>

**Re: simplify a trig expression***From:*"David W. Cantrell" <DWCantrell@sigmaxi.org>

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**Re: simplify a trig expression**