Integrity is the alignment between what you say and what you do. Trust is the belief someone has of your integrity.
The way to develop integrity is to watch what you say and do what you said you'll do. Watching what one says is hard, so it's easier to under-promise or under-state. Doing what one says is also hard, so the most important thing is to execute just the key things that you think makes the biggest difference.
Trust is slightly harder because it's the perception of you from someone else, and thus not directly under your control. I think the key to engender trust is to be consistent; that is to demonstrate integrity across time and under different conditions. When that happens people tend to be comfortable with your character.