Integral $\int_0^1x^{\frac{1}{x}}{\rm d}x$

Question

I've been trying to work out $\int_{0}^{1}x^{x^{x^{.^{.^{.}}}}} {\rm d}x$ and by using integration by inversion, I've managed to get it down to $1-\int_{0}^{1} x^{\frac{1}{x}} {\rm d}x$, and I was wondering if there's any way to solve this, either as an infinite series or in closed form? Possibly in terms of the $\Gamma$ function? I'm very stuck

Thanks in advance

Answer 1

I think I now have worked out why I was having such issues. As pointed out by @user121049 the power tower stated in my question is only defined from $e^{-e}$ to $e^{\frac{1}{e}}$, and so the integral is undefined between 0 and 1. As such, below is my solution for the integral of the same curve, but over the portion on which it is defined:

$\int_{e^{-e}}^{e^{\frac{1}{e}}}x^{x^{x^{.^{.^{.}}}}} {\rm d}x = e^{\frac{1}{e}+1}-e^{-e-1}-\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\sum_{c=0}^{a}\frac{c^b(e^{-c}+(-1)^{a+b}e^c}{c!(a-c)!\prod_{d=1}^{b+1}(a+d)} \approx 1.24$

Thanks so much for all your help.