Is this Non-Inverting integrator correct?
I think your math is correct. However, real op amps will not work with this configuration, because real op amps have finite frequency response, or a time delay, or they shift the phase of a sine wave: all of which are about the same thing. 

So, when you try this with a real op amp, the circuit will be unstable. 

You can see this mathematically by adding a pole to the gain of the op amp... perhaps writing it is A/(s+a), or A/s. (s being the Laplace variable, "a" a small number),   then doing the math in the Laplace domain.    

This just shows that using an ideal model can lead to the wrong conclusions sometimes.