One reason for broadening is chemical exchange.
For example, if the proton you are referring to is in exchange with another population of protons, for example from a solvent, the integrated area could represent the total population that is exchanging.
Coordination of a paramagnetic metal ion is another source of broadening, but integrated. Integrated area should be unaffected if there is sufficient time between each scan. However, since the paramagnetic metal ion can cause faster relaxation of nearby protons, it is possible that the integration is distorted by protons near the paramagnetic ion fully (or closer to fully) relaxing between scans, while the other protons do not. This effect is often used intentionally to enhance detection of protons near a paramagnetic metal center.