What is the integration of $\int x^{|x|} dx$?
Actually through several google search finally I have found a solution for the problem $\int x^{x} dx$ using Gamma function function and I am hardly sure that the same method can be applied to solve the problem if $x^{x}$ is going to be replaced by $x^{|x|}$.
So, what is the appropriate approach to solve the problem $\int x^{|x|} dx$
On
assuming that this is for $x\in\mathbb{R}$ you will still run into problems with the output of the function being complex for $x<0$ since you will have a negative number to a non-integer power, so in that respect this function is very similar to the functions $x^x$ and $x^{-x}$ so as others have suggested there are many papers on that integral so I suggest that you use those in this manner: $$\int x^{|x|}dx=\begin{cases}\int x^xdx&x>0\\\int x^{-x}dx&x<0\end{cases}$$ as for at $x=0$, both cases will return the same result as it can be proved that $\lim_{x\to 0}x^x=1$
$x^x$ and $x^{-x}$ are not integrable in elementary terms. See This theorem by Liouville. This theorem one way to approach that question.
This is obviously not something one would expect, and I too was surprised at this when I first asked myself your question.
It's possible (and a good exercise if you're interested) to find a closed form expression for the area under $x^x$ and $x^{-x}$ over the interval $[0,1]$. Spoilers:
$$\int_0^1x^xdx=\sum_{n=1}^{\infty}\frac{(-1)^n}{n^n}$$
Once you've done that, you can do essentially the same for $x^{-x}$. This, as far as I know, is the closest one could come to an answer to your question.