Integra DTR 7.7 or Denon AVR-3806

There is not enough difference in power to make it a real concern, I would look more into features you desire and other issues such as ease, remote, set-up. If you trust your dealers opinion then go with Integra, both have XM ready radio service, HDMI 1.1, and MM Phono.
I think Integra is better in that it supports Dolby 7.1ES while Denon doesnt in that model. Also the INtegra has up to 240 watts per channel of dynamic power in the front 2 channels so it can handle more then it looks like on paper.
Having owned the Onkyo (Integra) products and being familiar with friends Denon units.....I would go Integra.

If, like 99% of the people out there, you have serious acoustical issues, associated with typically small domestic spaces/rooms, YOU NEED THE DENON!!!! The room correction/EQ (audyssey MultEQ) is simply going to help bring the overall sound quality to another level compared to the Integra...regardless of the inherent sound potential of that unit! Basically, the benefits of helping to fix the acoustic nightmares (that brings the sound quality of even the most expensive equipment money can buy down to low/mid fidelity) in the room and system, which the Denon can do, FAR OUTWEIGHS any relatively small preferences for one units inherent sound quality over the other!
Infact, if I was interested in the unit with the best sonic potential, all things considered, I'd go with separates, every time. However, comparing receivers, the only reason I'd get one, is if it could help add to the equation with some sort of advanced room correction and EQ'ing,...which the Denon does. In a small room, especially, the integra based system, alone, WILL HAVE YOUR SYSTEM SOUNDING LIKE A LOW-FIDELITY BOOM BOX!!!! The Denon in that same room/system will sound absolutely fabulous for a receiver!!!! Infact, with Multeq engaged and balanced, you'll never go back I think. The benefits are that much!
Others who know nothing, nor consider anthing in regards to the acoustical limitations and challenges in their room's, will likely differ. However, having sold the Denon 5805, B&K 507, and Arcam AVR100 and such, I personally would never chose ANY receiver for my system, if it didn't offer more than just what a basic "purish" sounding receiver could offer on it's own, sonically. And again, on their own, receivers are s "performance compromise" in terms of dyanmics, channel separation, signal to noise, "pre-amp section", refinement, etc, as it is. That said, I wan't all the help I can get in the areas that count the most, if I must use a receiver to drive a budget system, personally. And that means I wan't what the Audyssey can offer acoustically. Just one perspective...