"How can I show that I am free?"
Translation:Come faccio a dimostrare di essere libero?
I'm pretty sure you use dimostrate to demonstrate something, mostrare to show something (e.g. "look what I found!") and indicare to point something out (e.g. "show me the one you want.") English allows "show" for all three, but I don't believe Italian does. So, no, I don't think you can use mostrare here, but maybe a native speaker will chime in.
I think it's complicated - see http://italian.about.com/od/verbs/a/aa031908a.htm. It appears that the use of 'di' or 'a' is associated more with the preceding verb than the infinitive.
I also came across another comment from an Italian speaker that proposes a more general rule (see http://answers.yahoo.com/question/index?qid=20130906004535AA9sS0r), but I can't really see how to map it easily onto the list of verbs from the first link.
Finally, here's a link that suggests that it's easiest to memorize a shorter list of verbs that take 'a', while most of the others take 'di': http://tutorino.ca/grammatica/2007/10/26/uses-of-the-infinitive-in-italian.html (though 'fare' is not on the list :().