Expectation-Maximization-Algorithm: example exponential family

Hi everyone!
I’m in desperate need of an example (one with actual numbers!) for the EM-algorithm concerning exponential families. I spend a lot of time trying to find one, but unfortunately didn’t succeed. The only one I found, is a very easy one, that I unfortunately don’t fully understand. It’s at the very end of this site (where x=5 and y is unknown):

I think I do understand most of the EM theoryand know how to derive the log-likelihood function. I also understand how to theoretically “build” the E-step, but have no idea how to get to the solution (for the E-step) stated on the site. In my opinion, I would have to use integration in parts, but whatever I do, I do not get the same solution.
I also do not know, how they managed to get the stated iteration formula in the M-step from maximizing the above term.

Any help would be very much appreciated, as I really cannot find any better example!
Thanks a lot!