Sometimes the Professor will say things quickly, or say things before I can write them down. It would be of great benefit if I could record audio for the course, but I am not sure the legal guidelines around doing so.
The legal issues involved will, of course, depend on what legal jurisdiction you are asking about. However, there are some common factors that can be considered.
In the UK many institutions take the interpretation of the Disability Discrimination Act (DDA) that teaching staff cannot prevent students making private recordings (Audio or Video) of any teaching activity. The reasoning is that students are entitled to keep their condition confidential (as it may be a private medical matter) but also must be permitted to receive the appropriate adjustment for their condition. It would be expected, as a courtesy, that the student would inform the staff concerned that recording was taking place, but that this could not be enforced. (*)
Conversely, the Intellectual Property (IP) of the class and the material contained therein would be held by the University and the individual teacher to varying degrees. The recording could not be published or used for any means other than private study as this would then violate the rights of the owner of the IP.
Many aspects of disability discrimination, privacy and intellectual property legislation is common across many EU states, but there are variations in practice. I can imagine how these different statutes might interact in (say) the US, but I suspect it might vary from state to state and not always be a federal matter, and so I decline to speculate.
(*) I have responsibilities for assisting academic staff understand these issues in my department...
The first step should be to ask your professor. This is not just a common courtesy. If s/he is OK with it, great. If not, then you can probably gauge from the answer what the concerns are. That will show you how you might go forward. Often professors make available powerpoint presentations or other such visual aids. Perhaps it is a matter of the professor not speaking clearly or loudly enough, and sometimes they are not even aware of the problem. Etc.
What you should not do: just turn on your recording device without fair warning. This may set you up for all sorts of misunderstandings and repercussions. Don't go there.
In the US, allowing such recordings would probably be in the realm of "reasonable accomodation" as required under ADA. To claim such an accomodation, you would have to follow the procedure and have the Disability coordinator intervene, and you would have to have a disability (so the inconvenience of having to ask for a repetition would not be sufficient).
When it comes to copyright, you can't copyright an idea, only the expression of one, so it could depend on whether the professor had put the lecture in writing or other fixed form. If the professor has written his lectures down, they have fixed form and are protected by copyright law. If he lectures extemporaneously, the copyright is held jointly between the professor and the person who first fixes that form of expression. (I recognize that that has ludicrous implications, but I didn't write the law. It would be unwise to test the copyrightable status of unwritten lectures). In addition, contract law may supercede copyright considerations, the relevant question being whether the university has a policy allowing or prohibiting such recordings. Universities might make money from recording and selling access to lectures, so they might have an economic interest in forbidding recording.
Unilateral recording potentially conflicts with the privacy rights of students and the professor in the course. The major concern is that recording lectures can have a chilling effect on students' willingness to speak in class (hence the need to obtain permission from all of the students in the class, in Missouri). In some states (two-party consent states) you would have to obtain consent from all parties in a conversation: you could legally test whether a lecture with or without questions from the audience constitutes a "conversation". The problem is that if you're not in a two-party consent state, then you're in a one-party consent state, which means that you need consent from one of the parties -- and if it's just the professor talking, he may be the only "party" (it is not clear whether just showing up makes you a "party" to a conversation). Also, bear in mind that wiretap laws are potentially applicable because they don't exclusively refer to speech over wires. I would suggest getting the expert opinions of a panel of lawyers.