appears one of my textbook
s (H Priestley, Introduction to integration
) under the wonderfully descriptive heading:
Non-integrable functions which can be integrated
What Priestley's trying to say is that the function is not Lebesgue integrable, because the absolute value |sin(x)|/|x| is not integral, but the improper Riemann integral of sin(x)/x between 0 and infinity exists, and can be shown by some kind of analytic witchcraft ("an easy exercise") to equal pi/2.
So that's one for the hordes of you who thinks Lebesgue integration is the best thing since sliced bread. Priestley reckons it is a "small price" to pay, but I say Riemann integration gets the kudos for this one.