What is the process for finding the integral of y=x^x and why is it difficult?

[Vorde]

When we say that we cannot express $\int$x^xdx in terms of elementary functions, what do we mean by that?

Is it that y=x^x cannot be integrated, or that we cannot find it's integral, or is it something else?

 

[micromass]

It means that it's integral exists, but we can't write it down. That is: if we have all the well-known functions like +,-,*,/,exponentiation, logarithms, sines, tangents,etc. at our disposal, then we still couldn't solve that integral.
We can only solve that integral by inventing a new function.

 

[Vorde]

Thank you, that was confusing me a bit.

 

[JJacquelin]

In addition to what micromass already said and for more information about the integral of x^x :
The paper "Sophomores Dream Function"
http://www.scribd.com/JJacquelin/documents

 

[kdbnlin78]

Could approximate-around x=0 x^x looks like;

x+x^2 ((log(x))/2-1/4)+1/54 x^3 (9 log^2(x)-6 log(x)+2)+1/768 x^4 (32 log^3(x)-24 log^2(x)+12 log(x)-3)+(x^5 (625 log^4(x)-500 log^3(x)+300 log^2(x)-120 log(x)+24))/75000+(x^6 (324 log^5(x)-270 log^4(x)+180 log^3(x)-90 log^2(x)+30 log(x)-5))/233280+O(x^7)+constant

Then integrate and feed the result into Mathematica/Wolfram alpha - you may find it looks very complicated but at least one can write it down.