how to integrate e^(e^x) ??

Question

Could you please help in solving this?

Answer 1

If you make the substitution z = e^x, then dz = e^x*dx, dz = z*dx, so dx = dz/z and the integral becomes

∫(e^z)/z dz,

which from http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions

is

ln|z| + ∑(z^i)/(i*i!)..... [i from 1 -> ∞]

ln(e^x) + ∑((e^x)^i)/(i*i!)..... [i from 1 -> ∞]

x + ∑[e^(i*x)]/(i*i!).... [i from 1 -> ∞]

Answer 2

Start with e^z and z = e^x and just use the chain rule.

Answer 3

Chain rule

Dx (e^(e^x)) = (e^x) (e^(e^x)(1))