Should I write a "proof" to a professor I don't know? TL;DR

I solved two integrals left unsolved in a paper I'm reading. Should I notify the author? 



I am studying in a finance master and we have been assigned a project for one of our class. This project consisted in some literature review and then a small implementation.

Reading one of the papers, in which the author propose a new density p  to model a random variable, there is written that 


They can't find analytically the constant c that makes p a density (basically could not compute the integral of p) 
They write that numerical simulations show that E[X]= theta but an analytical proof was still missing


I have calculated in closed form both integrals, and numerical simulations suggest that my results are correct; I will probably add this to the project I'm writing.

Now the question is: should I send an email to the author about this? If so, how to phrase such an interaction? I guess that I'm looking for something along the lines of "thank you, well done" with maybe a short mention if they get around to modify the original paper. 

On the other hand, the paper is old (2007) so I don't even know if the author is still interested in that.

Also, the proof was nothing too complicated. I mean I spent a lot of time on it (mainly because of my inexperience) but it certainly doesn't feel like anything too difficult. Probably I just "saw" the right path to do it and once you see that, it's pretty easy.

Is this something common to do? Are professors pleased to receive such an email or just annoyed?
The significance of your additional work is unclear. Have you talked about this with the professor that teaches your class? (s)he – or another prof in the department – should be involved if you want to take this further.

If the additional work is significant enough, you'll be perfectly within your rights to write and publish your own paper that presents the results. Trivial results usually aren't publishable on their own, but if it's a novel approach then it's certainly doable. The original work is old enough that you need not worry about "scooping" the original authors' results. Just make sure to check that a solution to the problem hasn't already been published elsewhere.

In any other case, there shouldn't be any harm in sending an email to the authors (whether or not they will respond is another matter). Don't send the solution in your first email! Just inform the authors that you think that you've solved the problem, and only send them your solution if they agree to let you co-author any publications that incorporate your results. Intellectual property rights are a tricky business, and you should always ask for the help of a professor when doing things like this. The paper's authors will also be much more likely to take notice if the first email is from a well known professor, rather than an unknown grad student.