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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!

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up vote 2 down vote accepted

No, not Calculus AB level. This antiderivative is "not elementary" in the technical meaning of that term.


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

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Then could you provide a non-elementary solution? I also have a friend who is interested in this. – Ovi May 16 '13 at 16:20
This is related to one of the "non-elementary" functions which is defined by its integral, called the exponential integral Ei(x). Here's the result for $ \ a^{a^x} \ $ : [Sorry... should have figured there'd be more answers coming...] – RecklessReckoner May 16 '13 at 16:26
It's ok, thank you very much! – Ovi May 16 '13 at 16:33

$$\int e^{e^x}dx$$ 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 $\displaystyle \int_{-\infty}^{x} \frac{e^t}{t} dt$.


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