example for W|Alpha (and most CAS's)
Bruce Yoshiwara (guest) 17 Jun 2009 20:49
If you enter
1/(2+cos(x))
at W|Alpha, you get a couple of things worthy of discussion. The first is that both graphs appear to drop below the x-axis (which would be surprising for a function that is always positive), until you notice that the horizontal line showing the scale is not the x-axis at all but y = 0.4.
The fun part for Calculus is that, because the given continuous function is positive everywhere, its anti-derivative is increasing (and differentiable) for all values of x. However, the given expression for the indefinite integral (in W|Alpha or any other CAS using the Risch algorithm) is periodic: f(x + 2pi) = f(x).
And no continuous periodic function is increasing everywhere.
