Finding $\int x^xdx$

I’m trying to find $\int x^x \, dx$, but the only thing I know how to do is this:

Let $u=x^x$.

\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &=\frac{x^{2x}}{2} \end{align}

But it’s certain that this isn’t the correct way to evaluate that, and the answer must be wrong.

$$\int{x^xdx} = \int{e^{\ln x^x}dx} = \int{\sum_{k=0}^{\infty}\frac{x^k\ln^k x}{k!}}dx$$