Question

Why we dont add arbitrary constant in case of integrating factor in differential equations??

Answer 1

Because the integrating factor is just something you're using as a tool to solve it, sure, there may be an infinite number of different choices you could pick, but you might as well pick the easiest one!

Answer 2

Well, you can if you want; they'll cancel out.

Answer 3

If your differential equation depended on which integrating factor you picked to solve it, then that would mean it wasn't a legitimate method in the first place. It's sort of like, if I walked to the store to get bananas or I drove my car to the store to get bananas. It doesn't really matter how you get there, as long as you're getting the bananas.