This is an older thread. The previous responces are correct for older versions of iOS, but iOS 11 has built in screen recording, including audio. It appears to work with FaceTime, but I have not tested it with an actual call.
First you have to add it to your Control Center in Settings > Control Center > Cutomise.
To use it, pull up the Control Center and tap the icon. You get a 3 second count down before recording starts. Tap and hold the icon in Control Center to toggle the microphone on/off.
The status bar turns red to let you know you are recording.
To stop, pull up the Control Center again and tap the record icon. Yes, all videos will end with the Control Center panel. You can edit it out easy enough later.
The videos are saved in the Photos app; where you can do some basic editing.