[math-fun] Mysterious but well-known identity

This following identity has been floating around for years: integral_0^1 1/x^x dx = sum{k=1,oo} 1/k^k It's reasonably straightforward to prove, but the usual proof (which I'll leave as an exercise for anyone who hasn't seen it), though short, is not especially enlightening. Is there some "explanation" for the form of this identity (*) integral_0^1 f(x) dx = sum{k=1,oo} f(k) (where f is analytic), or is this merely a mathematical "coincidence" (which if it had no proof, might not have a proof) ? --Dan