Not quite. Be careful with units. The bucket loses 0.2 pounds per second, not per foot, so the equation gives . But now the problem is that we have the variable time inside an integral with dx in feet, so we need to relate time and distance traveled. We know the bucket moves at 2 ft/s, so you have x = 2t, or rearranging: t = 2/x. Now we can sub that into the integral:
Hope this helps.