How to prove that $\int_0^1 x^{-x}\text{d}x=\sum_{n=1}^{\infty} n^{-n}$

Question

I've been trying to prove that $$ \int_0^1 x^{-x}\text{d}x=\sum_{n=1}^{\infty} n^{-n} $$ Mathematica told me that's true, but I don't know how to solve it. Is there any way to prove it?