Minecraft is not too difficult to run on a laptop with integrated graphics from what I have seen on youtube.
I recommend you look into Lenovo ThinkPad E series business laptops like the 15.6" E550 ($550). They come standard with the Haswell generation Core i3-4005u CPU with the Intel HD 4400 graphics core which is fine for playing Minecraft @ 1366x768 resolution. But they offer a free upgrade to the new Broadwell generation Core i3-5005u with the Intel HD 5500 graphics core which is estimated for the moment to be equal to the Intel HD 4600 for better graphics performance. All you need to do is select the Core i3-5005u as an upgrade option. It has a HDMI port, and VGA port. I would have recommended the 14" E450 ($579) but to make the laptop thin and light it has no DVD drive. It also does not have a VGA port. The E550 only comes with 4GB of RAM, it can be upgrade to 8GB for $80 or you can buy a stick of 4GB of RAM for $40 or less and install it yourself. Just make sure it is 4GB PC3 DDR3L-12800 1600 1.35v RAM.
Alternatively, there is the ThinkPad E555 ($455) which uses an AMD APU instead of an Intel CPU. It comes standard with the A6-7000, but I recommend upgrading it to the more powerful A8-7100 APU for a total cost of $484. While the A8-7100 is a quad core CPU, it is no more powerful than a Haswell generation Core i3 CPU. The integrated Radeon R5 (Kaveri) graphics core is estimated to be about equal to the Intel HD 4600 / HD 5500 graphic cores. No game benchmarks for the Radeon R5 (Kaveri) and Intel HD 5500 graphic cores are available yet to the best of my knowledge. Again, it only comes with 4GB of RAM so upgrading to 8GB is generally recommended.
http
/shop.lenovo.com/us/en/laptops/thinkpad/e-series/?menu-id=thinkpad_e
The E555 ($484) works out to be less expensive compared to the E550 ($550). Overall in terms of processing and graphics performance they probably about the same overall. The big difference would be battery life. Lenovo claims 9 hours for the E550, but only 6.3 hours for the E555. Of course that really depends on what you do. Intel's Broadwell generation CPUs uses less power than AMD's Kaveri generation APUs (like the A8-7100) which explains the difference in battery life.
Although the product page states a 1080p screen upgrade is available for the ThinkPad E555, it is not an upgrade option yet. The 1080p upgrade is available for the ThinkPad E550 though for $80. From a productivity point of view I would definitely upgrade the laptop to the 1080p screen. Playing Minecraft at 1080p with the Intel HD 5500 graphics core should still be playable. You can always lower the resolution or install an optimization mod for Minecraft called Optifine to play the game.
https/optifine.net/home
You can click on the following link to Minecraft being played on the Sony Vaio Duo 13 with the Intel HD 4400 graphics core (less powerful than the Intel HD 4600 / HD 5500) @ 1080p. I do not believe the Optifine mod is used.
https/www.youtube.com/watch?v=PQjfTzpqG90
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