The vSphere Integration collects task and events from vCenter Server, but not the low-level internal logs from each vCenter Server component. These logs are used by the vSphere Content Pack.
Configuration for vCenter Server 6.5 and later releases should be done through the vCenter Server Appliance Management Interface. For more information about how to forward log events from vCenter Server, see vSphere documentation about redirecting vCenter Server Appliance log files to another machine.
For earlier versions of vSphere, although the vCenter Server Appliance does contain a syslog daemon that could be used to route logs, the preferred method is to install a vRealize Log Insight agent.
For information about installing vRealize Log Insight agents, see Working with vRealize Log Insight Agents.
The vSphere content pack contains agent groups defining specific log files to collect from vCenter Server installations. The configuration is visible at https://LogInsightServerFqdnOrIP/contentpack?contentPackId=com.vmware.vsphere.
For information about working with agent groups, see Working with Agent Groups
For information about vCenter Server log file locations, see http://kb.vmware.com/kb/1021804 and http://kb.vmware.com/kb/1021806.