As an integrated address is different from a subaddress and account hierarchy, how do I get the balance of an integrated address?
An integrated address is just a normal address with an embedded payment ID. Thus you use the usual get_balance RPC method, or
balance detail CLI command.
As they have a payment ID, you can also make use of get_payments (RPC) and
payments <PID> (CLI), which will show you any incoming payments made to the address using the provided payment ID.
It's worth pointing out you should not use integrated addresses / payment IDs, as these are deprecated in favor of using subaddresses.