Can I add uncooked noodles directly to soup? Most noodle soup recipes that I see call for boiling and draining the noodles separately, then adding them to the broth already cooked. Is there any reason that I can't just add the dry noodles 10 minutes before the soup is done, and add a little extra water or broth to compensate for that the noodles take? Doing it this way would allow the noodles to take up some of the broth's flavor, and also save dirtying an extra pot and a colander.
similar to what julio said, the main reason is that typical wheat-based noodles release a lot of starch into the water, which changes the consistency of the soup. the starches can add a dirty colour to the water, but more than that, they can also thicken the soup undesirably (think of making a roux). Finally, if there are leftovers, the noodles can sometimes become completely water logged, making eating it the next day a bit of a soggy experience. 

That said, with all these considerations there are times when you might add the noodles directly in-- non-starchy noodles (eg. like rice noodles) seem to do ok. also parboiling regular pasta before hand seems to help a lot with the starch and can still help you achieve the flavour integration you mentioned.