# Expectation Maximization for a 2D Normal Model

I'm working through an example in Richard Duda's Pattern Classification on Expecation Maximization Algorithm. Specifically I'm trying to understand the expectation part, and how the parameters get estimated via integration.

Below is the example:

So far so good. However I can not seem to derive the same equation below.

My question is how does it take the integral and get the equation in the second image, for the parameter vector theta?