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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](http://math.stackexchange.com/faq#dontask).2012-06-13
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    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 For example, my version of Mathematica (outdated ver 3.0) gives $$ \int_{-2}^{-a}\frac{1}x\,dx=-\pi i-\ln 2+\ln(-a).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?2012-06-13
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    I don't see how this question can get answered like others on this site.2012-06-13
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    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.2012-06-13

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