Refactor Riemann sum integral sensor to prepare for time based trigger (#113932)

* Refactor Integration sensor.

* Use local simple function to verify the State is numeric.

* Merge two methods to one.

* Method renaming: _handle_state_change

* Move async_write_ha_state to the caller.

* Add comment on why attr_icon is set to None

* Remove possible None type of State in validation methods.

* Use a dict to map method name to method class.

* Explain derived unit after integration.

* Renaming to _multiply_unit_with_time and elaborate in docstring.

* Set integral unit_of_measurement explicitly to None if source unit_of_measurement is None

* One function for unit of measurement related steps.

* Improve docstring of _multiply_unit_with_time

Co-authored-by: Erik Montnemery <erik@montnemery.com>

* Apply f-string suggestions from code review

Co-authored-by: Erik Montnemery <erik@montnemery.com>

* Be more clear in comment about removing the sensors icon default.

* Apply suggestions from code review

Co-authored-by: Diogo Gomes <diogogomes@gmail.com>

* Update homeassistant/components/integration/sensor.py

* Update homeassistant/components/integration/sensor.py

* Update homeassistant/components/integration/sensor.py

---------

Co-authored-by: Erik Montnemery <erik@montnemery.com>
Co-authored-by: Diogo Gomes <diogogomes@gmail.com>

tests/components/integration/test_sensor.py

@@ -563,7 +563,7 @@ async def test_units(hass: HomeAssistant) -> None:
 
     # When source state goes to None / Unknown, expect an early exit without
     # changes to the state or unit_of_measurement
-    hass.states.async_set(entity_id, None, None)
+    hass.states.async_set(entity_id, None, {"unit_of_measurement": UnitOfPower.WATT})
     await hass.async_block_till_done()
 
     new_state = hass.states.get("sensor.integration")