Hi gablu, and welcome to WebmasterWorld!
You've only done step two of a two-step process...
You need to edit your pages --or edit the script or database used to produce your pages-- to correct your URLs.
The links on your Web pages define your URLs, and mod_rewrite cannot "change" them.
Once you have edited the links on your pages, the rule you posted (with the modification & correction shown below) above will "re-connect" requests for the new
URLs to the same
server filepaths that used to be used to serve requests for your old URLs.
If your "case studies" are always numeric, and if there is never any additional information in the URL following the case study number, then I would suggest changing your pattern to be more-specific:
RewriteRule ^casestudy/([0-9]+)$ /casestudy_details.php?id=$1 [QSA,L]
Note also that the [R] flag has been removed, as this is an internal URL-to-filepath rewrite, not a redirect.
Once this is working there is an optional third step, which can be used to speed up search engines' removal of your old URLS and replacement with you new URLs. But that is all you can use it to do, it does not "change the URLs." We can discuss that once you get the first two steps done.
For more information on this subject, see the thread titled
Changing Dynamic URLs to Static URLs [webmasterworld.com] in our Apache Forum Library.
Jim