From the book he co-authored with his wife How to Enjoy Writing: A Book of Aid and Comfort.
Integrity, is, to me, a somewhat stronger word than “honesty.” “Honesty” often implies truth-telling and little more, but “integrity” implies wholeness, soundness, a complex philosophy of life.
To have integrity is to stand by your word, to have a sense of honor, to do what you have agreed to do and to do it as best as you can. To have integrity is to be satisfied with nothing less than the best job you can do.
In that sense, anyone can have integrity, regardless of how small and unimportant a role he may play in the world…
A bit later, Asimov gives a short example of his concept of integrity:
Integrity not only simplifies your life by making it easy to come to a decision, but it may keep you out of trouble.
A writer I knew slightly once suggested that I write a book very quickly and that I then engage in complicated financial dealings that would involve my risking some money to begin with. The book I wrote would, however, fail and that would enable me to write off so much money as a loss that I would save on taxes many, many times what I had invested in the book. Of course, we would have to be certain that my book would be a failure, so I would have to undertake to write a really bad one. I would be taking advantage of a “tax shelter” in this way, and it was all perfectly legal.
I shook my head. “No,” I said. “It’s perfectly possible for me to write a bad book while I am trying honestly to write a good one, but writing a bad one on purpose is more than I can undertake to do, no matter how much money it would save me on taxes and no matter how legal it might be.”
I walked away and, a couple of years later, I read that the fellow who had advanced this proposition to me was now on trial for this same “tax shelter,” I was rather relieved that I had been simpleminded enough to have integrity.