How can you read a high current from the integrator using A Rogowski coil ?

The Rogowski coil that I have has no details and it is air-cored coaxial coil. So what should I do with it?

Sorry, run a known AC through a conductor which has the rogowski coil wrapped around it. Measure the output of the coil to derive a calibration factor.

Put the output of the coil into an integrator - an op-amp will do nicely. I'd put something on the input to clip the input signal to the supply rails, just in case the output  from the coil is too high, but that is VERY unlikely.