Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
FYI, the formula is explained in – 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
up vote 2 down vote accepted

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

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


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

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

share|cite|improve this answer
@ 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.