The most difficult part of your question is obviously the first because most teachers and students wholly associate Atticus with honor and integrity. For many readers, even the examples below might not register as Atticus being disingenuous.
Lack of integrity:
1. After Miss Caroline tells Scout not to read at home anymore, Atticus tells Scout that she can read at home but that she must not let her teacher know. Of course, Miss Caroline's command is absurd, but for a parent to tell his child to disobey her teacher and to appear to be obedient might cause some to question Atticus's integrity.
2. Secondly, Atticus never tells his children that he is such a good shot and makes it appear that he is completely unathletic and unskilled when it comes to physical activities.
3. Finally, Atticus agrees with Heck Tate that no one should know that Boo Radley is the one who killed Bob Ewell. He does so to protect Boo.
Now for the easier part! Examples of Atticus's integrity:
1. The most signficant example is that Atticus takes Tom's case so that he will be an example to his children and live what he always tries to tellthem. His decision not only demonstrates integrity to his children, but it also allows Atticus to be true to himself. He mentions that he could never hold his head high if he didn't defend Tom.
2. Atticus is extremely honest with Helen Robinson. He does not sugarcoat Tom's situation, and when he must complete the difficult task of telling Helen that Tom has been killed, he does it himself, rather than passing it off to Calpurnia or someone else. This example expresses once again that Atticus--as Miss Maudie explains--takes on the tasks that no one else in town wants to complete.
3. Ironically enough, the end of the novel, while showing an intentional lie on Atticus's part (see #3 above), also shows to what extreme he will go when it comes to encouraging his children to act right and honestly. When Atticus thinks that Jem has killed Bob Ewell while defending Scout, he wants the truth to be known and for Jem to face whatever he needs to face in order for Jem to practice honesty.
In all of these examples, "good" and "bad," Atticus's actions are based on what he thinks is best for others. Even when he omits the truth or "bends" it, he does so for the sake of his children.