repairing concrete patio flooring & attached walls
The slab has a structural issue with the 1/2" rise this tells me that you have had a frost heave (if you live in the north), poor compaction under the slab originally or someone drove something heavy on it.
Any repair you do will at some point in the future crack/heave again.
Unlike commercial/industrial projects, residential slabs have the least care taken to ensure items like this don't happen. Inspections are not as strict so this has been happening for years and will continue for years to come.
The best solution pending budget is to remove and replace the slab, this involves chipping/cutting out the old, proper subgrade prep and compaction, forming, rebar mat, repour, cut expansion joints to help with normal/natural cracking of concrete.
Putting a cap in what you already have is possible but will always crack where the existing cracks/joints are and will again heave/settle where you have the current issue.
A DIY can complete the prep work with a little research, I suggest a qualified concrete contractor to do the pour and finish, that is the easiest part of the job, but if done incorrectly looks horrible.