I like integrated amps so have looked around at quite a few... not many come with a sub out. Also, they are not optimal for surround setup as they are typically just 2-channel/stereo sound. If an integrated amp has 4 channels, it's usually because it has A/B channels, and hooking 4 speakers up and playing A/B concurrently in the same room is just quadraphonic sound (not surround).
That said, there are integrated amps with "Pre-out" outputs which can be used for sub-out. The Marantz PM8005 is a great integrated amp, but runs for $1199. It has A/B channels, many output/input options, and tone control. It is also pretty powerful for integrated amps.
There is also the Maverick Tubemagic A1. This one is minimalist compared to the Marantz. It only has one stereo channel, a single analog input (besides line-in), and a single pre-amp out. It also pushes less power than the Marantz, but is still great for decently sensitive speakers. The A1 starts at $199. MavAudio also makes a Tubemagic A2 which is a class A tube amp with the same function as the A1. The A2 is not listed on their site, so compare to the very similar APPJ PA0901A tube amp.
The Nuforce DDA120 is different from the two above because it features a DAC.
There are 2 options for hooking up a subwoofer to the integrated amp:
Your best bet is to run the integrated amp's speaker-out to the speaker-in on the subwoofer. Then, send the speaker-out on the subwoofer to your L/R speakers. Your powered subwoofer most likely has a low-frequency bypass/crossover that filters the low frequencies to the sub, and the higher frequencies to the speakers. This route is required for the APPJ PA0901A as it does not have any outputs other than the speaker outputs.
You could also run the integrated amp's speaker-outs directly to your speakers, then use the pre-out on the integrated amp to the stereo-input on the subwoofer. This method would not take advantage of the low-frequency bypass/crossover function.