While carpooling is widely adopted for long travels, it is by construction inefficient for daily commuting, where it is difficult to match drivers and riders, sharing similar origin, destination and time. To overcome this limitation, we present an Integrated system, which integrates carpooling into transit, in the line of the philosophy of Mobility as a Service. Carpooling acts as feeder to transit and transit stations act as consolidation points, where trips of riders and drivers meet, increasing potential matching. We present algorithms to construct multimodal rider trips (including transit and carpooling legs) and driver detours. Simulation shows that our Integrated system increases transit ridership and reduces auto-dependency, with respect to current practice, in which carpooling and transit are operated separately. Indeed, the Integrated system decreases the number of riders who are left with no feasible travel option and would thus be forced to use private cars. The simulation code is available as open source.