How can I calculate $\int\limits_{0}^{\infty}e^{-e^x}dx$?

Question

I have no idea to start this.

Any hint or solution would be highly appreciated.

Answer 1

The hint is that you (probably) can't do this without using the exponential integral (a function):

$$\operatorname{Ei}(x) = -\int_{-x}^\infty {e^{-t}\over t}dt$$

With that function you could solve it if you do the substitution $u=e^x$.