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

[Alejandroman8]

how integrate x^x^x ?my teacher ask me)what do you think about it?

 

[HallsofIvy]

I would suggest numerically! Certainly there is no elementary function that has that as its derivative.

 

[Grufey]

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.

 

[Alejandroman8]

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

 

[JJacquelin]

@ 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)