Unfortunately there is no way to install proprietary drivers without risking changes to xorg.conf . Trying to use your integrated graphics will only complicate things further. You may want to try a headless linux / server edition and just forget about the integrated graphics if mining is really your main goal. Unity is known to take its toll on light weight gpus, if you want a desktop you might try Lubuntu then driver install, or start with server linux, install gfx driver, then install a desktop.
Making a backup of xorg.conf before this is done can save a little trouble in some cases. Check out section 4.2 in this link below for a step by step.
https://help.ubuntu.com/community/BinaryDriverHowto/AMD
Edit- even though your computer did not need a xorg.conf on basic install, you often need to add one to gain more control over gpus.
These examples are using an Nvidia gpu to generate a xorg.conf, however I see nothing that would stop it from working with AMD/ATI (except for the one answer that calls for nvidia-xconfig)
Instead "sudo amdconfig --adapter=all --initial" can be used as a replacement for that command to generate a xorg.conf for multiple gpus.
How to make an xorg.conf file
Full documentation can be found here https://wiki.ubuntu.com/X/Config
This includes some docs on dual gpu systems with hybrid graphics like yours.
There are no cryptocurrencies left that will generate anything but a loss in electrical usage when mined with a gpu though due to asics being the large part of the scene, if it is merely for testing and curiosity of how it works by all means plug away, you may want to visit the bitcoin.stackexchange for some further advice.