Show that is an integral representation of Euler's constant.
To justify the first change of order, you probably could write the integral as . Then change the order of integration, integrate with respect to x, and then justify bringing the limit inside of the integral.
For the second change, perhaps you can write it as where b is value such that the integrand is now always positive. Then do the same thing as above.
I think in both cases you could use use dominated convergence theorem to justify moving the integral inside of the integral. I would really like someone to tell me if I'm completely off base or not. This has been drving me a bit crazy over the past few days.