Final answer:
The term for the integration of lysogenic viral DNA into the host genome is a 'prophage', part of the lysogenic cycle where the virus becomes latent within the host cell and can alter the host's phenotype, a phenomenon known as lysogenic conversion.
Step-by-step explanation:
The term used to refer to the integration of lysogenic viral DNA into the host genome is termed a prophage. During the lysogenic cycle, a temperate phage's DNA is incorporated into the bacterial chromosome, at which point the phage genome is referred to as a prophage. The bacterial host containing a prophage is called a lysogen. When the bacterium replicates, it also duplicates the prophage's DNA, passing it to its offspring. This cycle allows the viral genome to persist in the bacterial population over time. Induction, which is a different process, happens when the prophage DNA is subsequently excised from the bacterial chromosome, which can lead to the lytic cycle where the phage replicates and lyses the host cell. Importantly, the presence of a prophage can lead to lysogenic conversion, which can alter the phenotype of the host bacterium, sometimes making it more virulent if virulence factors are carried by the prophage.