How does Atticus show integrity in To Kill a Mockingbird in reference to the quote below?
"I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It's when you know you're licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do."
Atticus demonstrates his integrity without flaw, throughout To Kill a Mockingbird. In this quote from Chapter 11, he is talking about Mrs. Dubose. Despite the fact that she's cranky, generally unliked, and has even made racist comments about Atticus' involvement with Tom Robinson, Atticus considers what life must be like for her. He never assumes. And here, he doesn't simply assume that Mrs. Dubose is just a mean, hateful woman. For whatever reason, Atticus is privy to the information that Mrs. Dubose had been addicted to morphine (perhaps as a result of some illness, likely the one that killed her). Mrs. Dubose knew her death was approaching but she wanted to kick her morphine addiction before she died, to "die free" as Jem put it.
Atticus notes that he and Mrs. Dubose did not agree on many issues, but she demonstrated courage in doing what she thought was right. Near the end of Chapter 11, Atticus says, "According to her views, she died beholden to nothing and nobody."
This description of Mrs. Dubose is analogous to the trial. Atticus knew it would be difficult, if not impossible, to get Tom a verdict of "not guilty." But, he did his best anyway. This is how he demonstrates his integrity. He notes that courage is doing the right thing even in spite of the likelihood that you will fail and in spite of the possibility that no one will notice your efforts. This may sound redundant but what Atticus means is that courage is doing the right thing for no other reason than it is the right thing to do: not for reward or approval.