I discovered a new analytical integral. How should I present this knowledge?
You actually have three choices: physics, pure math, and applied math. It might be useful in any of those domains.
But there are two questions to ask, and they are related.

In which domain is this result most useful?
How is it connected to other things within each of those domains?

A pure mathematician will find the result most interesting if you have strong connections. A physicist or applied mathematician will find it most interesting if it is useful outside its original application.
That's not to say, of course, that any of them would find it uninteresting. But they are all looking for things that can be used to extend their own work.
So, I would try to figure out where it fits most naturally and write it up accordingly, focusing on either the connections or the usefulness. Then send it to any good journal in that domain and listen to the feedback you get. Reviewers may agree that you have chosen correctly or not.