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2.2 GHZ processor with an integrated graphics card with a cap of 64MB[but I've pushed it to 250]

It's an older laptop from 2010 so that might be my problem.

It might be your integrated graphics card that's causing you trouble.

Though in saying that, the only requirements that the engine itself specifies (according to Steam) is Windows 7 & OpenGL 2.1 or newer

I'd say, make sure Windows is up to date & check what version of OpenGL you have, then try running the game again and see how that goes