You could upgrade to a first-gen i5. Those have been discontinued for a while so you would be shopping for either for a used part or a somewhat rare inventory leftover.
If you want to upgrade to current i5/i7 models (LGA1155), you will need a new MB.
But it says, "supporting the latest intel® core™ i7, core™ i5 and core™ i3 processors featuring the first time ever integrated graphics on chip and lga 1156 socket(core™ i5 and core™ i3 processors only), gigabyte h5x series motherboards set a new standard in multimedia platform for high definition contents playback. with the gpu integrated directly into the processor die, gigabyte h5x motherboards support 2 channel ddr3 memory, wide range of display interfaces that includes vga, dvi, hdmi and displayport (depends on model). additionally, the gigabyte h5x series are optimized for greater graphics performance by enabling overclocking the gpu core frequency for a performance gain of up to 13%."