TITLE: Integrating $\exp (\exp (x))$
QUESTION [9 upvotes]: How am I able to integrate $e^{e^x}$?
$$\int e^{e^x}dx$$
Am I suppose to use $u$ substitution?  But what should I let $x$ be?  And what should $dx$ be?
Thanks for the help!

REPLY [15 votes]: The answer will not be given in terms of elementary functions. If we denote $u=e^x$, then $dx=u^{-1}du$, and so your integral becomes the so-called exponential integral
$$\int \frac{e^udu}{u}=\mathrm{Ei}(u).$$