The phrase "integrity" refers to a person's level of honesty, moral convictions, and readiness to do the right thing. Honesty would be one of the components. For example, we want our doctors to be open and honest with us about diagnoses, not to try to prescribe drugs that we don't need, and to act in our best interests. We expect this since physicians are often regarded as persons of great integrity with strong moral compass in most communities.
What ‘integrity’ means and how it is expressed and perceived, depends overall on the practices of its stakeholders and their beliefs. From a social viewpoint, it can often be observed that it is a dynamic concept, so its precise meaning changes both over time and by context. That's because it results from implicit or explicit sensemaking processes among the stakeholders involved. So of course, integrity is always subjective.
The participle in this sentence is "sweating". It's a participle of the verb " to sweat" and it's a present participle: which we know because it ends with an -ing (past participles end with "-ed).
