Now I am the Ignorant Youth

A group of faculty in my department are discussing (over an email list that I am on) the weak integration skills possessed by students who have completed our calculus sequence. One example of this was that few of our students can integrate $\int \frac{dx}{2 + \tan x}$.

I was unconvinced by that datum. I don't really see the inability to evaluate $\int \frac{dx}{2 + \tan x}$ as sufficient for declaring a crisis of integration.

Now, if the claim had been that students enter the calculus sequence with such gaps in their background knowledge of algebra and trigonometry that they couldn't follow a step-by-step solution to $\int \frac{dx}{2 + \tan x}$, I'd be more interested in the debate. When we have students in Calc 3 who believe that the square root is linear, I don't think that "integration" is where we need to be focusing our attention.

How would I evaluate $\int \frac{dx}{2 + \tan x}$? If my computer was handy, I'd use the integrator. If not, I'd see if there was someone within shouting distance who knew what the trick was. After that, I'd resort to the usual algebra fussing and trig indenties. Only after that would I realize which substitution to make.