Are you running the BC file included in the SQL 2008 media? If you are then it would have been better to have included backward compatibility as one of the features in the SQL install. (Hindsight is wonderful...)
The BC component can be requested for both 32-bit and 64-bit installs of SQL Server 2008, but is only honoured when you do a 64-bit install. If you install on a 32-bit OS then you have to manually run the BC install.
If you are trying to use the SQL 2005 BC media for a SQL 2008 instance this may be the reason it is failing. Look for the SQL2005_BC.msi file in the SQL 2008 media \x64\Setup\x64 folder and run that.
The DTS Designer component is only available in the SQL 2005 Feature Pack, but this version works OK on SQL 2008 (and SQL 2008 R2).
Original author: SQL Server FineBuild 1-click install and best practice configuration of SQL Server 2017 2016, 2014, 2012, 2008 R2, 2008 and 2005. 1 Dec 2016
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Disclaimer: All information provided is a personal opinion that may not match reality.
Quote: "When I give food to the poor they call me a saint. When I ask why they are poor they call me a communist." - Archbishop Hélder Câmara