# Definite integral of tetration between $0$ and $1$

In my old writes I found next formula, where is ${_{}^2}x$ is tetration:

$$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$

And now I am interested in series of generalized case of tetration:

$$\int_0^1 {_{}^n}x \ dx = ?$$

Could anybody find out it with explanation?

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FYI, the formula is explained in en.wikipedia.org/wiki/Sophomore%27s_dream. –  KennyTM Oct 2 '12 at 11:15
I just proved a generalization of the Sophomore's Dream for equation $(2)$. Sophomore's Dream, as cited above, uses $q=-1$. –  robjohn Oct 2 '12 at 16:23

Let $n \in \mathbb{Z}^+$, $x>0$ and

$$a_{n,k}= \begin{cases} 1 & \quad \text{if k=0}\\ \dfrac{1}{k!} & \quad \text{if n=1}\\ \displaystyle \frac{1}{k}\sum_{j=1}^k ja_{n,k-j}a_{n-1,j-1} & \quad \text{otherwise.}\\ \end{cases}$$

Then

$$\int {}^n x\, dx= \sum_{k=0}^n \frac{(-1)^k (k+1)^{k-1}\Gamma(k+1, -\log x)}{k!} + \sum_{k=n+1}^\infty (-1)^k a_{n,k} \Gamma(k+1, -\log x) + C.$$

Source: I.N. Galidakis, On an Application of Lambert’s W Function to Infinite Exponentials, Corollary 10.9.

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@ Argon : Darn I was about to post that. Its one of my favorite formulas. –  mick Oct 2 '12 at 21:34
It's great, but what ablout case of definite integral inside of $[0; 1]$ interval? –  KvanTTT Oct 3 '12 at 4:46
@KvanTTT Find $F(b)-F(a)$? –  Argon Oct 3 '12 at 19:42
Yes. Did you surprise because of this series expansion is trivial for you? :) –  KvanTTT Oct 3 '12 at 19:51