I've corrected the model link.
I'm also posting links for the correct battery and for the replacement guide. It's a fairly easy swap, but it does require a specialized tool: the Y1 tri-wing screwdriver. Apple uses these weirdo screws to hold your battery to the computer chassis, in order to prevent you from doing exactly what you're about to do.
The tri-wing screwdriver is available from iFixit as a standalone screwdriver, and as a bit in their screwdriver kits.
MacBook Pro 15" Unibody Mid 2010 Battery Replacement
MacBook Pro 15" Unibody (Mid 2009-Mid 2010) Replacement Battery
Tri-point Y1 Screwdriver
Essential Electronics Toolkit
54 Bit Driver Kit
Classic Pro Tech Toolkit
4mm Screwdriver Bits
The Mid 2009 and Mid 2010 generations use the same battery.
For future reference: The model number isn't a great way to ID your computer. The model ID (A1286, in your case) identifies the form factor, rather than a specific generation of the model. A1286 is the model number for every 15" Unibody MacBook Pro (October 2008-October 2013), regardless of the specific features of any of the seven generations inside that form factor case.
A better method for hardware ID is to use the model identifier, which appears in the format (ComputerModelName)X,X. A Mid 2009 MacBook Pro would be IDed as MacBookPro5,3; your Mid 2010 is MacBookPro6,2. You can locate your model identifier from the About This Mac screen by clicking the System Report button (10.9, 10.10) or the More Info... button (10.8 and earlier). This will open System Profiler, an Apple utility that gives a comprehensive description of the hardware and installed software on your system. The model identifier is a few lines down in the Hardware tab, which is the first screen you see when System Profiler launches.