TL;DR: Whenever I use an OpenGL 3.1 application, the 3D part of it is black, because my  Mobile Intel HD/Radeon HD 6630M (apparently) uses OpenGL 2.1.

Before you say anything, I bought this laptop for studying (and the graphic card system is shit).
I have a Lenovo Thinkpad Edge E520, and it uses 2 graphic cards (check image attached), with the Intel HD as primary.
I cannot upgrade my Catalyst from 11.5 to 11.6. It forces me to get drivers from lenovo.com, which isn't up to date.
I cannot change my graphic card, I tried everything (yes I tried BIOS too).

Any help or advice would be much appreciated!
I can't play Amnesia/BRINK/Space Engine due to this, and most games have fucked up graphic glitches (Trackmania, Magicka, etc.)