^{x}does not have an analytical integral. You can try to do it by hand, or you can get a computer to do it, but you won't get it.

I've always been curious about the x

^{x}function. After finding that the LambertW could give me its inverse, I started wondering about its integral. Although I can't get it in f(x) form, I can do the next best thing, which is graph it (first plot, compared with x^x). It diverges at a slower rate than x

^{x}, which is to be expected when you look at its derivatives.

Since d

^{n}x/dx

^{n}of x

^{x}is x

^{x}is x

^{x}(1+ln(x))

^{n}, I figured that the integral would similar to the function DIVIDED by 1+ln(x), which is a reasonable guess at high x, but falls apart for low x. The second graph is the ratio of the actual integral to my guess.

Anyway, we can use this thread for talking about one of my favourite functions, x^x.