TITLE: Integral involving inverse of $x^x$
QUESTION [5 upvotes]: My brother gave me the following problem:
Let $f:[1;\infty)\to[1;\infty)$ be such that for $x≥1$ we have $f(x)=y$ where $y$ is the unique solution of $y^y=x$. Then calculate:
$$
\int_0^e f(e^x)dx
$$
I couldn't figure out something helpful, so any help is highly appreciated. I think he has the problem from some contest, but  I don't know, from which.

REPLY [5 votes]: Substitute $x=u\ln u$.  Note that $1\ln 1 = 0$ and $e\ln e = e$.  Thus
$$
\int_0^e f(e^x) \,dx
= \int_1^e f(e^{u\ln u}) (1+\ln u)\,du
= \int_1^e u(1+\ln u)\,du \\
= \int_1^e (u + u\ln u)\,du
= \left[\tfrac14 u^2 + \tfrac12 u^2\ln u\right]_1^e
= \tfrac34 e^2 - \tfrac14
$$