How do I record FaceTime
If you have a Mac you can connect your iPad to, it's fairly straightforward to record the iPad's video output in Quicktime -- although you might just as well use Quicktime to screen record from the Mac's copy of FaceTime in that case.

Last I checked, Apple still doesn't allow general-purpose screen recording apps in the app store (apps in the Games category can do it, but not elsewhere).

There are a few apps which claim to allow you to do screen recordings on the device without jailbreaking, but all the ones I've seen install via alternative mechanisms and so 'buyer beware', as it were. I haven't tested personally.

Edited to add:

@Allan To screen record your iOS device on a Mac is pretty straightforward, provided you have the necessary software versions (iOS 8 or later, macOS Yosemite/10.10 or later):


Connect your iOS device to your Mac using a Lightning cable 
Open Quicktime (player) on your Mac and from the File menu, select
"New Movie Recording".
A window will appear, which will likely default to showing you the
live view from your webcam. Click the dropdown button by the red
record button, and in the list of sources you'll see the name of your
iOS device (e.g. Derek's iPhone) listed under both the Camera and
Microphone sections of the input sources list. Select it in both
places.
You should now see your iOS device screen mirrored on the Mac (you
may need to unlock your device and 'trust' the Mac if you've not
connected before). 
Press the record button in Quicktime to get started.


Hope that helps.