Area under $\frac{1}{x^x}$ curve

Question

How can I calculate area under $\frac{1}{x^x}$ on any interval, I tried the Archimedes method, but I get

$$\frac1n\sum \frac 1{X_n^{X_n} }$$

and that's very complex to calculate because of the roots, is there an easier method to calculate this?

Answer 1

Sorry for the short answer, but here is the link with my work. It needs a better form though. You can also use the definition of the Riemann Sum to find another value.

(My Answer)

Using the definition of the Riemann Sum, we can rewrite this as:

$$\mathrm{\int_{\Bbb R^+}x^{-x}dx=\lim_{b,n\to \infty}\frac bn \sum_{k=0}^n\left(\frac{bk}{n}\right)^{-\left(\frac{bk}{n}\right)},n\gg b}$$

Here is proof of this result: Graph