An integral problem?

Question

How do you integrate $e^{e^x}$? I was able to get it down to du/(ln u) but I wasn't able to go further. Thanks!

Answer 1

No, not Calculus AB level. This antiderivative is "not elementary" in the technical meaning of that term. https://en.wikipedia.org/wiki/Elementary_function

added

Maple says $$ \int \!{{\rm e}^{{{\rm e}^{x}}}}{dx}=-{\rm Ei}_1 \left(-{{\rm e}^{x}} \right) $$ where this "exponential integral" function is $$ \mathrm{Ei}_1(z) = \int_1^\infty\frac{e^{-tz}}{t}\;dt $$

Answer 2

$$\int e^{e^x}dx$$ Substitute $u = e^x$. Then $du = e^x dx$

$$\int \frac{e^u}{u} du = \operatorname{Ei}(u) + C = \operatorname{Ei}(e^x) + C$$

Where $\operatorname{Ei}$ denotes the Exponential integral $\displaystyle \int_{-\infty}^{x} \frac{e^t}{t} dt$.

See http://en.wikipedia.org/wiki/Exponential_integral