TITLE: What is the function $f(x)=x^x$ called? How do you integrate it?
QUESTION [15 upvotes]: For real numbers $x > 0$, the function $f(x)=x^x$ seems pretty cool. 
Is there a name for this function? It's obviously been studied before. 
It grows faster than exponential functions and factorials but slower than double exponentials. 

We can find its derivative by writing it as $$f(x) = x^x = e^{\ln x^x} = e^{x \ln x}$$
This lets us use the chain rule to get
$$f'(x) = e^{x \ln x} (\ln x + \frac{1}{x} \cdot x) =  x^x (1+ \ln x)$$
Since $x^x$ is never equal to zero, if we set $f'(x) =0$ we get $$1+ \ln x = 0$$
which gives
$$ x= e^{-1} = \frac{1}{e}$$

This (perhaps unsurprising?) appearance of $e$ might be enough to make this function interesting. At any rate, I tried to find its indefinite integral and failed miserably.
I guess my question is partially a reference request ... where can I read more about $x^x$?

REPLY [8 votes]: Assuming you know series, as the integral of that cannot be expressed in elementary functions(if you let any function then you can just define $\mathfrak{F}(x)=\int_0^x t^t\,\mathrm{d}t$), then;
$$\int x^x\mathrm{d}x=\int e^{x\log x}\mathrm{d}x=\int e^{\log x^x}\mathrm{d}x\;\;\stackrel{\text{series for }e^x}{=}\;\;\int \sum\limits_{n=1}^{\infty}\frac{x^n\log^n x}{n!}\mathrm{d}x$$