Mike, the documentation does touch on this. Under the Details and Options section of Plot reference (and the related Plot3D, ParametricPlot etc).

The wording can be seen as a bit cryptic. An alternative phrasing could be 'Since f will be evaluated at each plot point, it may in many cases be more efficient to use Evaluate[f] before specific ..'

Hans:

Thank you for the mind expanding insight.

From the documentation, I do not think I would have ever figured out that you could use Evaluate like this.

If uc[] is not constant, then I do not think that qMultiple1[] can be modified in such a way that Plot[] would be able to call it.

That said, you could define your own PlotQMultiple1[] that encapsulates the both the function and the integration.

Please note that I'm still relatively new here. So there are still things about the language that I do not fully understand. But, I learn a great deal from trying to answer questions like this. So, let me give this one a shot:

The reason that qMultiple2 is faster is that it is not recalculating the (close to the) same value (returned by the integral). This is a great optimization technique.

That said, I'm not convinced that the two functions return exactly the same thing. Just a quick look at the code and I see that uc[t3_] is defined with = and not :=.

When uc[t3] is defined in qMultiple1, = is probably what you want. But, when you pull the definition of uc[t3] outside of the qMultiple2, then I think it should be defined with :=. Otherwise, you are getting the same value for uc[] every time.