What effect will a virus in the lysogenic cycle have on an organism?

The correct answer is - integrate DNA of the virus in the host genome.  

Explanation:

The lysogenic cycle is a one out of the viral reproduction in which lysogeny leads to internalizing of phage into the genome of the host. In this process bacteriophage genome into the genome or formation of the genome of the host bacterium.

Virus or bacteriophage enters its's DNA or nucleic acid into the bacterium by membrane fusion or endocytosis. These can go in dormancy in or latency for a period of time and replicate itself in the host.

Thus, the correct answer is - integrate DNA of the virus in the host genome.