C#.net connect to sql server

You don't need to pass the username and password with SSPI - that's the whole point of it. :)

Windows knows who the user is (since they had to login to Windows to get to your app), and that's the username that will be passed automatically to SQL Server.  However, that means SQL Server will need to know who that person is -- i.e. there will need to be a SQL login for each user (or group of users) of your application.

You don't need to do anything different in C# to establish the db connection, as your connection string looks correct.