To my understanding E[X-1] = E[x] - 1. So why doesn't the following work?

A policy has a deductible of 1 and a capped payment of 5 and losses are distributed exponentially with mean 2. The expected loss in the range of 1-6 (where variable payment amounts are made) would be $\int\_{1}^{6}.5xe^{-.5x}dx$ which is about 1.42. I would have thought the expected payout was then .42 = E[x] - 1, but it's not. If you replace .5x by .5(x-1) in the integral you get roughly .86.

I see how the math works, but why doesn't using expectation as a linear operator work here?