TITLE: How are contributions to mathematics reviewed/published?
QUESTION [5 upvotes]: Let's say you're a hobbyist who's seemingly found a novel identity/solution/method or other general contribution to some field of mathematics. What you don't know however is:

If you're actually the first to discover it
If it is useful
If it isn't useful, is it still worth having reviewed?
How to have it reviewed
How protective should you be of your finding?
If it is useful, how do you ensure you're accredited for it?
If, how and when you should post it on stack exchange

While I am asking for generalized advice I'll offer some background for context: I'm an engineering student, and I enjoy trying to find solutions to non-elementary integrals via functional analysis, complex analysis, combinatorial methods, etc. simply because I like the challenge and I've learned some fascinating things about other fields of mathematics I wouldn't normally be exposed to. With that being said, I may have managed to find a solution to a couple of non-elementary integrals and while I'm aware they have limited application, I'd still like to know if my work is correct and worth sharing, and maybe even get credit if it would land me a scholarship or something. The last part may be simple wishful thinking, but who knows?

REPLY [6 votes]: Write it, submit it, and find out!
I'm in a similar situation - I am an amateur mathematician and I had an idea I wanted published, and was successful!  My paper was "Extending the Algebraic Manipulability of Differentials" and it was published in "Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis".
Here's the simple answer - write it up, submit it, and find out!  If it is rejected, they usually say why.  If you are part of an academic organization (i.e., a student), then I would find a professor who would help you initially, and then find the right journal.  Otherwise, I would write it up.
Another thing you can do is find a mathematician on UpWork.  For my own paper that's what I did.  Trying to get someone to give you worthwhile feedback is hard.  If you find a professional mathematician who is willing to look at it, chances are they are just going to be critical, because you probably didn't word things the right way, and mathematicians tend to be incredibly pedantic.  Paying someone is helpful because then they are trying to make you successful, and will help you fix it up and word it in a way that makes sense, rather than just telling you that you are "wrong" just because you don't have the precision in terminology that a professional would.
Also note - there are many good ideas which, while not new, certainly could use re-explaining or just reminding everyone of them.  You can often get these published as education papers.  That is, you aren't claiming that it is new, you are talking about math education and how it can be improved by focusing on this idea, or that the idea can be better explained to students by going a different route than usual.