Integrate x^x^x: What do you think?

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SUMMARY

The integration of the function x^x^x is complex and does not yield an elementary function. Participants in the discussion suggest numerical integration as a viable approach. The Lambert W function is introduced as a method to express x in terms of z, where x^x=z, leading to a more complicated integration process. Additionally, one user mentions that a primitive of x^(x^x) is represented by the function Sphd(1,1;x), referencing "The Somophore's Dream Function" for further insights.

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  • Familiarity with numerical integration techniques
  • Knowledge of advanced calculus concepts
  • Experience with special functions, specifically Sphd functions
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  • Research the properties and applications of the Lambert W function
  • Explore numerical integration methods for complex functions
  • Study the function Sphd(1,1;x) and its derivations
  • Read "The Somophore's Dream Function" for deeper insights into advanced integration techniques
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Alejandroman8
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how integrate x^x^x ?my teacher ask me)what do you think about it?
 
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I would suggest numerically! Certainly there is no elementary function that has that as its derivative.
 


I don't know, but I thougt in the Lambert function

If we define x^x=z, then, x=\frac{ln(z)}{W(ln(z)}, where W is the Lambert function. So we can write

x^z=\left(\frac{ln(z)}{W(ln(z)}\right)^z, and then integrate.

I know, that this is awfull, but, maybe it can help or give any clue.
 


Thanks)
but i think it took not one hour to take the result
 


@ Grufey : False because you forgot the dx.
So, even more complicated !
.
@ Alejandroman8 : a primitive of x^(x^x) is the function Sphd(1,1;x)
Is it a joke ? Just read the preamble of "The Somophore's Dream Function" :
http://www.scribd.com/JJacquelin/documents
A so simple answer ! (§.12) :smile:
 

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