A "family-integrated" church is one where children and babies are encouraged and welcome to stay with their family during the services, rather than be whisked off to a nursery or special children's program.
Our church would fit that label, though we don't really use it to refer to ourselves as that. To us, it's just the way things are, no label needed.
The older kids love taking care of Stephen during the service so much, they keep close tabs on whose turn it is next. Which of course also helps me out tremendously, as Stephen is at that active age where he wants to be all over the place, and eventually has to be taken to the parent/baby room every time.
Stephen just of late has discovered his love of "singing" along with everyone else, which is really cute. So while it lasts, it is very fun to have him sit with us and sing along. His attention span will grow as he ages. In my experience, babies up until about 9 months are very easy to keep in the service. From then on up until about 2 years they are a challenge, but beyond that are old enough to get the concept of sitting still and listening. And more importantly, learning.
Out of the mouth of babes and sucklings hast thou ordained strength because of thine enemies, that thou mightest still the enemy and the avenger. - Psalm 8:2