The problem in answering your question is that intensity itself can refer to many different things. From the context, I'm willing to say that the integrated intensity is the exposure of your sensor from a source that has a time-varying or otherwise non-constant irradiance.

My reasoning for this recommendation is based on the fact that for sources with constant irradiances, the exposure of the sensor is the irradiance of the source times the time that it had been observed.  This is the same as integrating the constant irradiance over time, so if you have a source with a non-constant irradiance, the integrated irradiance should provide a more accurate cumulative photometric measure than the traditional equation for exposure which again is irradiance times time.