Integrity in a relationship is always an intrinsic endeavor, really amounts to your self control. And it is definitely something to be proud of in yourself. I don't want to sound elite in this point of view however it is also a point of pride.
I am an older guy that has been in many relationships over my years and I am in my second marriage which is a fantastic relationship. Many times I have been tempted and have had many opportunities but I never cheated in any relationship and that is dear to me it sums up who I am and hope to be.
I guess what I am trying to say is it is up to you and only you. Either you are a one man woman when in a relationship or you are not. There is a lot to gain if you stay faithful to him and yourself or a lot to lose should you decide not to. Even if you do not get caught, you will know.