Final answer:
The digestive viscera's smooth muscle is largely innervated by the tenth cranial nerve, also known as the vague nerve. It plays a crucial role in the autonomic nervous system, affecting digestion through the rest-and-digest response.
Step-by-step explanation:
The smooth muscle of the digestive viscera is served largely by the tenth cranial nerve, which is also known as the vague nerve. This nerve is part of the autonomic nervous system, providing extrinsic innervation to the alimentary canal, including the smooth muscles involved in digestion. It is responsible for increasing gastrointestinal secretion and motility under the rest-and-digest response.