I didn’t click the link, but if it’s the very first teaser for the film (released in December of 2011 a full year in advance), I’m almost positive it can be heard in "Over Hill"
Yeah its the very first teaser & some of it is there. But I don’t think I heard the beginning & middle. Or its not entirely the same maybe
this isn’t a teaser but the actual trailer
"Over Hill" is the closest you’ll get since there is no "Misty Mountains" without the Dwarf Cast singing (half the song they are humming)
but the cue is featured in "Over Hill" in an orchestral manner like DAKoftheOTA stated earlier…
and btw there is no extended version of "Over Hill" 😛