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 try... 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"
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