Final answer:
A phage type that can integrate into the host genome and exist as a prophage during the lysogenic cycle is termed 'temperate'. These bacteriophages can bestow additional traits on their hosts through lysogenic conversion and can shift to the lytic cycle under environmental stress.
Step-by-step explanation:
The name given to a phage "type" which can incorporate itself into the host genome as a lysogenic prophage is a. Temperate.
When a temperate bacteriophage infects a bacterial cell, it may enter into a lysogenic cycle, where instead of causing immediate destruction through lysis, the virus integrates its DNA into the bacterial host’s genome, becoming a prophage. The host bacterium carrying this integrated viral DNA is dubbed a lysogen. The incorporation of the phage DNA provides the host with additional genes, which can sometimes enhance the bacterium’s phenotype, in a phenomenon known as lysogenic or phage conversion. This alteration can lead to increased bacterial virulence due to newly acquired toxin genes. An important example of such activity includes the lambda phage infecting Escherichia coli. The lysogenic state is generally stable, but under certain stress conditions such as starvation or chemical exposure, the prophage may be excised, prompting the phage to enter a lytic cycle, thereby producing new virions and lysing the host cell to spread the infection.