Someone has
integrity if they have unity of thought, speech, action, and feeling.
That is, there is no contradiction between what they are thinking, what they are saying, what they are doing, and how they are feeling.
This is stated in the
OT (Old Testament) and by Jesus in the
NT (New Testament).
Matthew 22:37 Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind. [kjv]
Having integrity in this sense is not sufficient.