I agree with others who have recommended OBS as a recording software. It's free, and once you get the hang of it, there's a lot you can do with it, especially easily switching between video recordings showing you talking, screen sharing, presentations, etc.
I also use additional equipment beyond my computer's built-in webcam and mic. I also agree with others that if you can only add one thing, get a better microphone. I use a Blue Yeti mic and the Elgato Facecam webcam.
If you want to do any editing of your video after the fact, I use the Wondershare Filmora software. It's cheap (but not free) and easy to use. I mostly use it for cutting out mistakes or re-recording a voiceover of parts that I want to change. But editing isn't necessary if you just want to record on OBS and directly upload that video to your LMS or a YouTube channel.
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Michelle Vered
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