A “simple” integral and the repeated logarithm

Extract

A question which I like to ask sixth-formers at their interview for University is to evaluate

not worrying for the moment about the behaviour at x = 0 or x = 1, or the limits of the integral. This question is difficult enough for some students but on a good day one can lead them through to the correct solution (log|log x|) by various suggestions like “try a substitution” (x = e^t) or “do you know a rule for when the top is the derivative of the bottom?”