An Integrated is a Tuner a PreAmp and an Amplifier all in one box (sort of like a Reciever but without the whole surround sound processing).Actually, an "integrated amp" is the pre/amp, amplifier and often a digital signal/surround processing facility, but without the tuner. (The tuner is what makes a combo box a "receiver".) As an example, my Yamaha DSP-A3090 integrated amp came without a tuner, but has a lot of DSP (digital signal processing) modes, such as "cathedral", "jazz club", etcetera, as well as Dolby Digital 5.1 processing.
However, the conclusion remains the same. You need a power amplifier if you are looking for more wattage to drive your speakers than is available via your receiver's built-in power amps.
One other concern -- does your particular model of receiver have "pre-outs"? That is, are there line level jacks on its back panel which allow connection of an external amplifier? Some models use these jacks with a "bridge" connection to input jacks for the built in power amps. If the "bridge" is removed, the internal power amps do not receive the signal from the receiver's pre-amp section. Others have the jacks for optional use, but the "bridging" is handled internally and in some designs it is "defeated" when the external pre-outs are used. If your receiver does have pre-outs, adding a power amp can take your system to a new level of performance without the expense of both a separate pre/pro and power amp all at once.
OTOH, if no such jacks exist on your receiver, the alternative of extracting the signal for the power amp from the receiver's speaker outputs is not the recommended way to pursue better performance, although it is technically possible with certain add-on devices. In the case of no pre-outs, it would be better either to step up to a more capable receiver or go with a separate pre/pro in addition to a power amp.