The music itself is copyright free (it was composed in 1801) so having the music itself is fine.
Where you have an issue is that each play of it is also copyrighted, this is due to the differences of each performance. This means that you have to be careful what performance you use. YouTube has a filter in its search for things that are creative commons, this means you can be more sure of the copyright status of the music, the full licence details of YouTube videos is in the more info section.
I am unsure of the copyright status of a digital performance (eg feed sheet music into a program and it plays the music) but most likely they get copyrighted upon first processing but subsequent processes of same data do not add to the copy protection
If you were to play it yourself or get someone to play it for you then you/they would have the copyright to that performance and therefor not have any issues with its use (assuming if you get someone else to play it you either get the copyright from them or a licence for the copyright)
I hope this helps