I'd say that context would determine what kind of adjective you might want for integrity. As an off-the-top-of-my-head definition, let's say integrity is a state of unity in which elements share a common denominator. Say architecture in Colonial Williamsburg here in Virginia. The restoration project includes preserving the integrity of the area as much as possible.

We could say that the well-integrated architecture of Colonial Williamsonburg serves as a fine example of successful restoration. I think I've heard the phrase "well-integrated" functioning as an adjective a lot.

Now when we speak of a person's possessing integrity, that's something a bit different. I don't think I've ever heard the phrase "well-integrated" applied to a person. Perhaps to a person's work, but not to a person possessing integrity.

Anyway, it'll be interesting to read whatever the rest of the word hounds write here.

Best regards,
WW