Integrity can be simplified to the foundational concept of being your word, and doing what you said you would do. If you said you would be at a meeting at 1pm, can people count on you to be there at 1pm? If you said you would contact someone to discuss something, did you contact them?
Over time, people have an expectation of you based on their past experiences with you. So, if you typically are late to meetings or appointments, then people expect you to be late. If you typically say you are going to make a phone call and then don't, people have an expectation that you won't have done it.

This time of year is often a good time to reflect. Is there anything that you want to change about how others experience you? And, if you think you haven't behaved as well as you would like in this area in the past, you can begin right away to change it. Just do one small thing that you said you would do, each day. In a short time, you will demonstrate to the others in your life that you can be counted on.
As the quote goes, any passing moment is a chance to turn it all around.