cheapest decent low powered integrated w/ sub out

Speaker level straight off of any integrated you like into the subs speaker level inputs.
Homerun of speaker wire from amp to main speakers without ever going through subs high-pass filter.
Input impedance of sub is so high that amplifier does not know it is there.

Good luck!

Bignerd100 is right, some high-end sub manufacturers like REL actually recommend this method over rca sub out. So you don't have to limit yourself to these conditions.

thank you both so much because that opens up so many more amps as options and just simplifies things generally...i imagine this is also more or less the case even if one is using a single driver crossover-less main speaker? (i'm not in search of sonic nirvana, just don't want to compromise any single excessively.)

thanks again for the education