How can I record synthesizer music to iPad?
As long as the synth you have is core MIDI compatible. You may use a Apple iPad Camera Connection Kit or the Apple USB 3 Camera Connection Kit if it requires a bit of power output.

Other alternatives to just straight up USB for non-core MIDI products (products that require drivers) include iRig accessories, or plugKEY. 

You will need this cable to make it work.

Best of Luck!