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Integral of e^(e^x)

Date: 12 May 1995 10:55:54 -0400
From: Andrew Ricker

What is the integral of e to e to the x power?

Date: 16 May 1995 22:21:01 -0400
From: Dr. Sydney

Dear Andrew, 

Good question!  When you say what is the integral of e to the e to the x
power, do you mean e^(e^x) or (e^e)^x?  I assume you mean the former because
the latter would follow the same pattern of what happens when you integrate
any constant raised to the x power  (If you are unclear on how to do this,
feel free to write back!).  

So, let's think about integrating e^(e^x).  Hmmm.....  Well, it seems to
defy all of the standard tricks and tools of integration like substitution,
trig substitution, parts, etc...  I couldn't find it in a table of
integrals (though I must admit I was skimming through).  So, how could we
approach this?  It seems to me that within limits, you could approximate
this integral with a Taylor expansion.  Do you know what Taylor polynomials
are?  If not, write back and we'll explain.  You could express e^(e^x) as a
Taylor polynomial and then integrate that polynomial (Polynomials are
certainly much easier to integrate!).  If you are looking for a nice, pat
answer, I don't think there is one.  But, you could give these polynomials a

If you have any other questions, or if you want to know more about what I
have said, feel free to write back. Thanks for the question!

--Sydney, "Dr. Math"
Associated Topics:
College Calculus

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