First, it is important to define integrity. If we define integrity as honesty with strong moral principles, then your short story should challenge this honesty and morality. That tension of temptation would be your main thrust.
In light of this, think about what your audience would find particularly tempting. Make that into the setting. For example, if your audience struggles with the love of money, think of a scenario where this person finds money that belongs to unjust bankers. If he could take it and no one else would know about it, would he? Would you?
You can even make the story even more complex. What if the person will use this money for a good purpose, such as giving it to an orphanage in a third world country. Does that justify the taking of the money? What would a man or woman of integrity do?