@Raven, keep in mind the standards for graphic design are different between the US and UK. In fact, the standards are different from college/university to college/university. Some graphic design courses can be passed with little more than photoshop, whilst others require software suites costing thousands for the "student" editions.
I do agree with you that integrated graphics are a waste of money for most graphics students.
@master00d, this may be nit-picking... but by definition, integrated graphics is not a graphics card. In fact, there are no graphics "cards" in laptops, which means replacing them is extremely difficult and, in all practicality, impossible. Laptops will have a graphics chip built into the motherboard. IGPs will have no memory of its own and instead shares it with the CPU. Core i3/5 will have the actual IGP built into the same chip as the CPU. Discreet graphics in laptops means it has a seperate chip/socket on the motherboard, usually withs its own exotic cooling system. This system is typically designed for one chip, so if you tried replacing it, you'll certainly face overheating issues.
I agree that 4GB of RAM is highly recommended for any graphics course, but that's cheap and easy to get/upgrade. The CPU itself isn't that important, since almost all of the work in graphics will be done with the GPU. However, the only way to get a 6 hour battery is to use a CULV laptop (very low powered CPUs), and Core i3/5 is NOT low powered (although the IGP does allow up to 4.5 hours).
If you did get a laptop with dual-graphics (probably with nVidia Optimus technology), the price would probably shoot above £600. Optimus doesn't come cheap. On the other hand, it does allow the IGP to be used for basic tasks (up to 3.5h battery) and discreet graphics for intensive tasks (like your graphics software). Just as an example, for the HP DV6T laptop you linked, it starts off at $600 and you need to add $100 to the base price for AMD HD6490M (and it doesn't mention Hybrid Graphics, which is the AMD version of Optimus).
Also, the US HP site doesn't help someone ordering for the UK. The nearest UK equivolent I can find is
this DV6 laptop. It has a HD5470 (512MB VRAM) and costs £450, so of all the suggestions so far, it's the best laptop for the needs of the OP. Personally I have some reservations about HP because of their frequent cooling problems and lack of customer service, but I'm still holding back from having a proper search of my own because I feel it's better if students seek out their own ideal laptop (with guidance) rather than have online people tell them what to get.
EDIT: @FAST, I think you may have too much faith in scholastic/academic computers. The few institutes I've visited (in the Greater London area at least, which includes parts of Surrey and Epsom) that actually have graphics cards in their computers (rather than IGPs) usually have serious problems with their network (and when the network goes down, or at least one of the servers for the intranet, many of the more powerful graphics software stops working). Even Epsom University, which is supposed to specialize in graphics, fine arts and fashion (and the source of the little I actually know about graphics design), encourages their students to work from their home computers because the ones they have frequently encounter "problems".