Welcome to Android Central! What do you mean by integrating private calendars? Are you talking about calendar events that were entered into the Phone account calendar (i.e., not saved to any of your Google account calendars)?
I have a number of my own Google calendars, as well as some calendars my wife shared with me, and a calendar from my daughter's school. It's very easy to toggle them on and off within the Google Calendar app, and to review, edit, or add new events. You have to make sure all of those calendars are properly synced, though. Go to the Google Calendar app's settings, tap on each of those calendars, and make sure the Sync switch is turned on.
04-16-2018 08:57 PM