There are any number of people who preach integrity; but most of them don't know what integrity means.
One definition is obeying the law and doing what one has to do. This definition is wrong. Doing what you have to do is not integrity; it is self-interest. It comes from anticipation of positive consequences and avoidance of negative consequences. This means that this does not qualify as integrity.
Another definition came from a columnist in Reader's Digest, who stated that integrity is acting as a single unit. This is a luxury of the ignorant. In an interconnected world, people will have all sort of influences, many of them incompatible with one another. Acting as a single unit is something that comes naturally when one has had only one set of influences. And that creates a perverse set of incentives - toward closed-mindedness and intellectual dishonesty.
What definition of integrity is most workable? The one I have found is this: Doing the right thing even if you don't have to do it. A man who takes care of his children even if he can get away with not doing it, or a man who chooses to work even if he can get away with not working, is a practicioner of real integrity. Such a person is doing the right thing even though he doesn't have to. And that puts him in a higher echelon than the people who do what they do because they must.
For this reason the prerequisite for all moral choice is this: Freedom. It is only within the context of freedom that a choice can be made based on its ethical merits rather than on anticipation of positive or negative consequences for oneself. The ideologies such as jihadist Islam, that seek to force morality, are in fact only forcing self-interested choices. And only the ideologies that affirm freedom have a possibility of arriving at true morality.