Question 1206915: find by steps: int (e ^(e ^(x))) from 0 to 1 dx
Answer by Shin123(626)   (Show Source):
You can put this solution on YOUR website! As far as I'm aware, there is no nice closed form for this integral. The best you can do is approximate it. You can of course just plug it into a calculator, but another way to do it by hand is use the Taylor series of 
The Taylor Series is  So for  , we get 
We can integrate each term separately.
It's not hard to take the integral of the individual terms, in fact, we can see a general pattern among the answers we get (except for the first one),
Again, there is no closed form for this, but we can get a good approximation by evaluating a finite amount of terms (the more, the better the approximation). For example, evaluating the first 10 terms gives about 6.315786, which is quite close to the actual answer of 6.316563.
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