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I think in precalculus students should be taught the following:

  1. Euler's identity for $e^{i \theta}$.
  2. The principal value of $log(x)$ for $x<0$.

Then in Calculus they should be taught that
$\int dx/x = \log(x)+C$ instead of $\log|x|+C$.

Likewise, teach them that
$\int dx \tan(x) = -\log(\cos(x))+C$

and so on. I don't think that would be too advanced. The advantage would be, that, what they learn will be consistent with what some will learn at a later date in complex analysis. Perhaps more important, students would get the expected answer from tools such as Wolfram Alpha and Mathematica. Any thoughts?

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"If your motivation for asking the question is “I would like to participate in a discussion about ______”, then you should not be asking here." --faq. –  Henning Makholm Jun 13 '12 at 11:25
Not all students will, sadly, ever take complex analysis. As they encounter this question for the first time in a calculus course, for their sake I vote for keeping the absolute values in place. The point about Mathematic/WA doing something is Wolfram's problem, not ours. For example, my version of Mathematica (outdated ver 3.0) gives $$ \int_{-2}^{-a}\frac{1}x\,dx=-\pi i-\ln 2+\ln(-a). $$ Is this really the ideal we should aim at? –  Jyrki Lahtonen Jun 13 '12 at 11:28
I don't see how this question can get answered like others on this site. –  Doug Spoonwood Jun 13 '12 at 11:28
I don't like the $\log |x|$ answer, but it is a fact that the student who learned $\log |x|$ in school make much less mistakes later than those who learned $\log x$ and remain totally unaware that there is a problem here. –  Phira Jun 13 '12 at 14:20
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closed as not a real question by leonbloy, Henning Makholm, Rahul, mixedmath, Jyrki Lahtonen Jun 13 '12 at 11:38

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1 Answer

While I like the sentiment, precalc students barely have any knowledge of complex numbers, at least where I am. If they won't understand the math then they shouldn't be taught it. If they do then they'll understand why the difference appears when they come to it.

Though I'd beg to differ about $\log|x|$ vs $\log(x)$. If anything they should be taught that the general antiderivative is

$\log(x)+C_1, x>0$

$\log(-x)+C_2, x<0$

Real analysis isn't complex analysis. While I would say that discrepancies like these should be delved into more closely in complex analysis courses than they sometimes are, I don't think it's appropriate to introduce concepts so far ahead of time.

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