I am trying to solve Exercise Problem 1.13 from Estrada and Kanwal’s A Distributional Approach to Asymptotics, Theory and Applications without any luck. Actually, I think that it is wrong.
Problem. Prove the following asymptotic expansion as $ x $ goes to $ \infty $: $$ \int_{a}^{x} t^{t} ~ \mathrm{d}{t} \sim x^{x} - x^{x - 1} \ln x + x^{x - 2} (\ln^{2} x - \ln x + 1) + \cdots. $$
My approach was integration by parts to find a recursion formula, but any way I do it, the first term that I get is always $ x^{x + 1} $, not $ x^{x} $ (that is why I think the problem could be wrong).
Can anyone point out the mistake, either from the book or mine? If I am wrong, what approach should I take?