haha integrated means that your graphics card is part of the motherboard. Which means I can't forsee any reason your PCI slots would be taken at all. All you will need to do is open up the case, pop in the card, and install the drivers.
It might require you to pull out some plastic or whatever, or break out some pieces of metal, but that's usually standard fare for custom builds too. New cases arrive all metaled up but pre-cut so you can usually bend the pieces out that you need to.
Like I said you need to make sure you have a power supply that will support a high power graphics card. For example my entire computer can run with only 300 watts, except for the fact that my graphics card alone uses 300 watts, so I have a 600 watt power supply (Or maybe it's higher, don't remember)
if not, usually either the computer wont turn on at all, or could possibly cause serious damage (Highly unlikely).
Here's a video on how to do it. It's from 2007, but the method and the mayhem are still basically the same.
This is a pretty good vid on helping you choose a card:
The last thing is, installing a game on a computer is as simple as putting the disc into your DVD drive
Usually it will auto-install.
Last suggestion, the top of the line cards from last generation, are now mega cheap. We're talking 75% discounts off things that used to be like 200-500 dollars. Or more. The top of the line back then, is still amazing these days.
The Last thing to remember:
OPENING YOUR COMPUTER WILL VOID ANY WARRANTY YOU MAY HAVE RECEIVED