I think in precalculus students should be taught the following:
- Euler's identity for $e^{i \theta}$.
- 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?
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.