Using a database (or an external connection to a service you are using) in an integration test is not only valid, but should be done. However, do not rely on integration tests heavily. Unit test every logic element you have and set up integration tests for certain flows.
Integration tests can be written in the same way, except (as you mentioned) they include more methods etc. In fact, the code snippet you've shown above is a common start write-up of an integration test.
