Implementing the I action in an analog PID controller
Your basic idea is right, but not the implementation.  Vin going to the wrong opamp input.  Vin should drive the + input.  The - input should be simply the output voltage fed back.

Otherwise, yes, this overall circuit will result in the negative integral of the input voltage.  The first opamp buffers Vin so that it doesn't have to be a low impedance source.

Note that the initial condition is not specified.  This is the voltage on C when this circuit is first started up.  If this is inside the feedback path, and some glitching at startup is acceptable, then this may be OK.