You can't integrate logic and emotion. Not if you expect the logic to work. However, there is no rule that says you can't be both a logical and emotional person. So long as you capable of making that distinction and understanding when it is time to use either.
Actually, you don't have to worry about when it is time to use your emotions because few people have any control over those. You just have to know when to use logic. If you are have that ability.
Logic is not a "sense." The sense is "reason." Logic is a process, naturally driven. That's physical nature, not human nature. And like most natural processes in reality, it depends on functions that are for the most part independent of human cognition.