The method I use (on a mac using ableton) is:
Install soundflower. Now you can set up soundflower as your audio output (in your daw). You can then select soundflower in your video software (I’ve always just used quicktime for the most part) as your audio source.
When it’s all set up right, just hit record and the audio is completely integrated with your video. Try it out.
If you are on a PC, search for similiar software to soundflower and you should be able to do it this way to.