I have the same problem. We have a SQL Server 2012 where the SSRS is installed and a separate SharePoint 2010 server. While installing SQL Server on the SQL box I did select both Reporting Services - SharePoint and Native but where I open Reporting Services Config Manager the SharePoint Integration Mode is not showing, I only have the native mode. When I try to run a new SSRS instance as a test, during the install test I can see the SharePoint Integration option which is ticked but greyed-out indicating that its already installed.

Does this mean on SQL 2012 if I want to run the SharePoint Integration Mode I have to install this on the SharePoint server?

Please assist.