The exact procedures may vary by airline, and I'm not aware of the $40 fee you reference, as while I've flown plenty of times with a firearm, it was always using either a dedicated rifle case, or by placing a pistol in my bow case as mentioned by
@Trial153. Note that the pistol in bow case approach is not without possible drawbacks. I've had a couple of my locks either cut off or caught in and destroyed in the machinery (I believe the former but was told the latter), and depending on the airport in question, you may not be able to be present when the case is opened for inspection. If your bow case has lots of points where locks can be used, I'd suggest putting a lock on each point. (Some TSA agents will insist on that, even though they are supposed to only require enough locked points to prevent prying open the case.)