From my experience I find a 22 cal rifle to be very effective. A shotgun tends to be too loud and you don't want to alert the neighbors. I usually have to shoot 5-7 beavers a year at my in laws place every year. I would say don't sit out to early as the beavers won't be out until its almost dark, then just blast away. If you shot them in the head while they are swimming you may have to be prepared to fish them out or else they will float to the surface and provide evidence of your misdeed. If they are in the irrigation channel they may float downstream away from you place if you are lucky. You can also wait until they are on-shore to shoot them too. Another method, but less humane, is to shoot them in the body so they swim away and die somewhere else.
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