Try that first. If it's still wrong, I'll look at the entire integral for you.
One other thing, your bottom limit should start at 0, or natural length of the spring. Just because you use the force given at 4in to calculate your k, does not mean that it automatically includes that in your answer. You must evaluate the entire interval from 0 to 6 inches (or in this ft-lb situation) 0 to .5ft