Find the cdf of the extreme-valued pdf $f(x)=e^{(x-e^x)}$

Question

Find the cumulative density function of the extreme-valued pdf $f(x)=e^{(x-e^x)}, x\in \mathbb{R}$.

I don't know how to integrate this or find the integration limits. I tried log transforming the pdf. I can't find a convergent integral.

Answer 1

If $u = e^x$ then $du = e^x dx$ hence $$ \int e^{x-e^x}dx = \int e^{-e^x}e^xdx = \int e^{-u}du. $$