Well, the Google "ad hoc grammar coach" does not necessarily mean that the right answer is the phrase that returns the biggest number of results, especially in combinations with prepositions. Usually these combinations have slight different meaning or even totally different meaning sometimes. So, "integrate into" (some people use "integrate in") means integrating objects or groups to be a part of a bigger object or a group. "Integrate with" means that 2 groups/objects are integrating with each other, like bringing them together. "Integrate" without preposition generally implies adding new members to a group (usually social). Not sure if "integrate to" is a correct phrase.