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

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Integrating the function x^x^x is complex, as there is no elementary function that serves as its derivative. A suggestion is to approach it numerically, while some participants discuss using the Lambert function for transformation. By defining x^x as z, the integration can be reformulated, but it remains complicated. The discussion also references a specific function, Sphd(1,1;x), as a potential primitive, but skepticism about its simplicity is expressed. Overall, the integration of x^x^x is acknowledged as a challenging problem with no straightforward solution.
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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