[RFC] Better integral docs

[lxvm]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• 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 SciML/SciMLExpectations.jl#148, and inadequate web documentation for the new interface, as in this issue SciML/Integrals.jl#218 and a discourse post. I want to address the following in this pr

• Restore the method to set inplace IntegralProblems for all integrands
• Publish the IntegralFunction and BatchIntegralFunction docstrings to the web documentation
• 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]

Codecov Report

All modified and coverable lines are covered by tests ✅

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

Additional details and impacted files

@@             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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