How to integrate this integral

Question

I am having trouble solving this (note: we have not studied it yet nor was Google of any help)

$$\int e^{x^x}\, dx$$

Answer 1

According to Liouville's theorem and Risch's algorithm, this primitive cannot be expressed in terms of elementary functions. Nor are there any special functions that I know of which can help express it either (by using some substitution, for instance). Not even the error function. Hope this helps.

Answer 2

$\int e^{x^x}~dx=\int\sum\limits_{m=0}^\infty\dfrac{x^{mx}}{m!}dx$

Then apply the approach similar to Series Expansion Of An Integral. and you will finally find that

$\int e^{x^x}~dx=\sum\limits_{m=0}^\infty\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^{n+k}m^nx^n(\ln x)^k}{m!k!(n+1)^{n-k+1}}+C$