# About the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$

I notice both wikipedia and mathworld have the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$ that:

$\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx=\sum\limits_{n=0}^m\dfrac{(-1)^n(n+1)^{n-1}}{n!}\Gamma(n+1,-\ln x)+\sum\limits_{n=m+1}^\infty(-1)^na_{mn}\Gamma(n+1,-\ln x)$ , where $a_{mn}=\begin{cases}1&\text{if}~n=0\\\dfrac{1}{n!}&\text{if}~m=1\\\dfrac{1}{n}\sum\limits_{j=1}^nja_{m,n-j}a_{m-1,j-1}&\text{otherwise} \end{cases}$

How does this result derived?

• With all due respect for tetration (very little, as far as I am concerned), this is one truly ugly result, imfho. I really hope they can find some good application of this, either in some applied or pure mathematics field, in some "real life" one or whatever, otherwise it is going to remain just an ugly result. The question though is interesting: how did they get it? +1 Apr 6, 2014 at 11:19
• I think this result is beautiful, not ugly. And I couldn't care less if it doesn't have any "applications". May 19, 2014 at 23:58