An integrator and LPF are the same thing.
In order to understand the link between a low pass filter and an integrator, you must understand what integration means in this context.
Integration is, in this case, the average signal level. This is in fact the DC bias level. So an lpf will in fact only leave us with the low frequency content, which will be close (ideally) to the DC level of the signal.
With a HPF we want the exact opossit. We want to get only the short-fast changes and discard the DC bias level. Showing the small differences and discarding the DC level.
This is exactly what an HPF is used for.