Use the SQL Server instance name instead of an IP. This will give you a connection string like:
Data Source=PCNAME\SqlExpress;Initial Catalog=NameOfDatabase;Integrated Security=True;Pooling=False
or similar.
Since you mention C#:
Try setting up a connection in VS with the Server Explorer pane:
1) Open Server Explorer.
2) Right click "Data connections" and select "Add connection"
3) In the dialog that follows, select your DataSource, and database, specify the security info, and press the "Test connection" button.
4) When the connection works, press "OK"
5) Highlight your database in the Server Explorer pane, and look at the Properties pane. A working example of the connection string will be shown, which you can copy and paste into your app or config file.