Often I've heard and cited Job's declaration, "til I die I will not remove mine integrity from me." But I never really looked at the context of the chapter, and assumed that Job was referring to the fact that he refused to renounce his faith in the face of his adversity. While that is true of Job, this particular declaration doesn't exactly come in that context.
In chapter 27, Job is in the midst of defending himself against his friends, who repeatedly insist that Job's woes must be a punishment from God and that he needs to repent. Job goes on, "My righteousness I hold fast, and will not let go: my heart shall not reproach me so long as I live." (Job 27:6) Job is saying that he won't remove his integrity in the sense that he won't give in to the temptation to second-guess his life, to fall into self-reproach, or to assume that God was punishing him when he had committed no great sin. He was whole through His redeemer, and his virtue endowed him with confidence.
I can certainly learn from that example, as confidence is not a strong suit of mine. Are false modestly, second-guessing, and questioning the trials I receive all forms of "remov[ing] mine integrity from me" or denying the Atonement in my life? Trials come in large part because that is what we signed up for when we came to Earth. Of course there are things that we should learn form our trials and ways that we should be refined, but if we are on the Savior's path we needn't feel that we are less that whole because our trials might seem punishing.