Screen Recording Program For iOS?

There is none without jailbreaking. Apple doesn't permit this sort of app in the App Store. There are hardware options that include connecting to your Mac/PC, but nothing directly on the device.
 
Quicktime lets you do it by default. Even lets you select an area.

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It's not a perfect solution, but you can use AirServer to turn your PC or Mac into an AirPlay receiver so you can mirror your iOS device's display on your PC/Mac and use whatever screen capture tool you like to record.
 
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