I use an app called Display Recorder but your iOS device must be jail broken and I use an iPad 4 which allows a fast recording rate. Apple will not give companies permission to offer apps that record the screen so you must Jailbreak your device and then purchase the app for $5 through the Cydia store.
Another alternative is to use a program called Reflector
http://www.reflectorapp.com/ that will allow you to use Airplay to a computer instead of an Apple TV and it allows you to record. I have never used it though and it cost $15 but looks cool.
Another program like the one above is Air Player.
http://www.airserver.com/ You can buy a single "commercial license" for $4 which is a cheaper alternative.
This page also explains a few other options.
http://www.makeuseof.com/tag/3-ways-...reen-activity/