[RFC] Better integral docs

[lxvm]

Checklist

• [x] Appropriate tests were added
• [x] Any code changes were done in a way that does not break public API
• [x] All documentation related to code changes were updated
• [x] The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• [x] Any new documentation only uses public API

Additional context

The updates to IntegralProblems in SciML v2 lost the ability to set an in-place IntegralProblem for a non-IntegralFunction, observed here https://github.com/SciML/SciMLExpectations.jl/pull/148, and inadequate web documentation for the new interface, as in this issue https://github.com/SciML/Integrals.jl/issues/218 and a discourse post. I want to address the following in this pr

• [x] Restore the method to set inplace IntegralProblems for all integrands
• [ ] Publish the IntegralFunction and BatchIntegralFunction docstrings to the web documentation
• [x] Improve the clarity of the IntegralFunction and BatchIntegralFunction docstrings

I would appreciate comments on my writing and how we should publish the docstrings. I will also open a pr to Integrals.jl to update the readme and documentation with examples of the new interface.

[codecov[bot]]

Codecov Report

All modified and coverable lines are covered by tests :white_check_mark:

Comparison is base (0c62711) 26.25% compared to head (26c20dc) 40.15%.

Additional details and impacted files

```diff @@ Coverage Diff @@ ## master #609 +/- ## =========================================== + Coverage 26.25% 40.15% +13.89% =========================================== Files 54 54 Lines 4113 4134 +21 =========================================== + Hits 1080 1660 +580 + Misses 3033 2474 -559 ```

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