means The quality of being honest and morally upright, holding firm to your principles even when no one's checking — and, in a related sense, the state of being whole and undivided.

from From Latin integritas, 'soundness, wholeness,' built on integer, 'untouched, whole, complete' — literally in- ('not') plus a root tied to tangere, 'to touch.' So integer means 'untouched,' unspoiled by tampering. That same integer gives mathematics its whole numbers, and it's why a thing with structural integrity hasn't cracked. The moral sense — a person undivided, all of one piece, not corrupted or broken into convenient parts — grew from that idea of wholeness.