The reason these were called integrity t-shirts was that they white with a blue 90's design and said the word INTEGRITY: DO THE RIGHT THING. Although it didn't seem that important back then, I am thankful that people in authority over me took the time to instill this principle and characteristic of Jesus.
Jesus is an amazing example of Integrity, because he did it right his whole life. The Bible says in Mark 12:14 (NIV), "Teacher, we know you are a man of integrity. You aren't swayed by men, because you pay no attention to who they are; but you teach the way of God in accordance with the truth." This powerful thought is displayed twofold from this passage of the Bible.
The first part is that this was at a point when the religious were trying to trap Jesus. Some of the religious leaders were testing Jesus about paying taxes. The amazing thing is that Jesus was calm, cool and collected in what could have been a time filled with turmoil, because of accusations. Jesus knew himself that he didn't have anything to hide from the beginning, so when people try to trap him he knew that he was doing the right thing. He was so calm the way I picture this is that he looked at the coin with Caesars face on it and in passing he flipped the coin to the religious people and said, "Give to Caesar what is Caesars and to God what is God's." Then walked away, while the on lookers were amazed in his calm integrity filled spirit. Jesus never had anything to hide, because he knew his moral compass.
The second part is that others knew of Jesus' integrity. Others knew that Jesus always did the right thing. This amazes me that not only people that followed him, knew he was a man of integrity, but also people that didn't follow him knew as well. It is one thing to have people that love you say you have integrity, but it is a whole different thing to have those not in your inner circle to claim the same statement. What a powerful expression by Jesus. This was something that didn't just take words but took actions as well.
The greatest action of integrity Jesus displayed was that to going to the cross. This isn't something that he was completely up for. While he was in the garden of Gethsemane in Matthew 26, He stated in verse 39, "My Father if it is possible, may this cup be taken from me. Yet not as I will, but as you will." Jesus didn't want to die, but knew that this was the right thing to do not for his own benefit, but the benefit of all of us. He did the right thing even though it meant death which was death on cross.
Since Jesus died for us, we have an amazing opportunity to live for him. I know that integrity is not the easiest thing to do, but it is the right thing to do. Whether that is apologizing to someone for doing/saying the wrong thing or acting in obedience when no one is looking. It is always honors the Father, when we choose to do the right thing. So I encourage you to walk in integrity and ask for forgiveness from others & the Father when you fall short, regardless you are doing the right thing.