Can I connect my integrated amplifier to a power amplifier and still use my integrated amplifier?

Your question is confusing because it is all in one paragraph. My response will be composed the way you should have composed the question. Compare the two and note the difference. Your idea should work IF the switching in the jacks allows it. However, it won't work if the switching does not allow it. Y adapters are cheap, so the financial risk is minimal. Nothing confusing about that, is there?

It could work but it also depends on what integrated amp your talking about and what power amp, also what speakers, when you split the signal like that there is no gate or regulator involved and the electricity can flow in any direction, what you should be concerned with is the impedance involved and how its perceived by each amp...this is going to determine the flow of electricity, The electricity will tend to take the path of least resistance...if all four speakers are the same make and model it should not be a problem because the impedance will be pretty much equal in all four channels but your results could vary depending on the amps too and how they were designed as compared to each other. If your power amp has 150 watts per channel while your integrated has only 40, this could cause your speaker out put to be mismatched with no way to regulate it, if you have different speakers attached this also could cause a mismatch in impedance causing one set to play really loud while the other you can barley hear, most amplifiers will have a trip switch or fuse that will protect the amp in case of to much flow, but sometimes you can also over heat an amp with too low of an impedance load, which may not cause the amp to fail immediately but could shorten the life expectancy of the amp from say 30 years down to 10......

I was taught that two amplifiers in series will always cancel out the first amplifier. So if you are trying to get more power to the speakers, you will not accomplish this with two amps, and a Y cable. Or are you trying to add speakers? you know what would help also , Give us the Model number of both devises so that we may see the outputs and connectors, it should be that you can take the pre out to "feed" a low output signal to the other power amp. What if you hook two pairs of speaker to each terminal? so if you have two speaker outs, and left and a right channel, hook took speaker the the left and two speaker to the right. Dropping the impedance to 4 ohms will not damage the amp, so your integrated amp, feed the signal to your power amp, and the power amp has four speakers on it! Problem solved!