Can the integral of $x^x$ be found? I'm interested in knowing if the indefinite integral of $x^x$ can be found in terms of elementary functions.
I am under the impression (be it correct or incorrect) that it can be found. This is why: the derivative of $x^x$ has $x^x$ in it ($d/dx[x^x] = x^x(\ln(x) + 1)$). The derivative is quite easy to find with logarithmic differentiation, but the integral – not so much. 
If the indefinite integral cannot be defined in terms of elementary functions, why? 
If the indefinite integral can be found, would you please work it out? I find myself lost here, I've tried multiple methods. 

Hint: $x^x=e^{x \ln x}=\sum_{k=0}^{\infty}\frac{(x \ln x)^k}{k!}$. Then interchange the integral sign and summation sign.
