I am not a good trapper. Oh I do catch them all the time in Cinnabears along their trails in and out of the river. But when they are wanting them removed for damage control I get permission from F&G to shoot them. I take a book, and sit out in the woods near where they have been working. If they have built a dam I will damage it so they will make repairs, then sit close to the area where I can get a good shot as they are working on the damage. It usually only takes a couple of days to totally remove them all.
I use either a .223 with Nosler Ballistic tips, or a .17 HMR. My partner uses .22 shorts. We try and catch them on land, it's easier that way. If they are in the water, it is another story, head shots are essential. An instant kill will result in a floating beaver. Anything else will result in a diving beaver, and they are gone for the time being. In quite water such as a small lake, or a pond, they will surface about two hours later. It takes two hours for the gasses to build and make them float. At that time a fishing rod with a Bass Plug, or Spoon is handy. Otherwise you need a canoe or boat.
In Alaska shooting is under methods and means of trapping. So it is legal to shoot any fur bearer, except Beaver. Therefore I have to get special permission to shoot them in the area I have been requested to remove them all.