If x+1/x is an integer then (x²+1)/x is an integer, but this can only be true when x=±1, because for all other x the result of the division (which is negative when x=-1) leaves a remainder of 1. When x=1, xⁿ=1, and when x=-1, xⁿ=-1 or 1. When n is even xⁿ=1 and 1/xⁿ=1, so xⁿ+1/xⁿ=2 and when n is odd, we have ±2 for x=±1. Therefore, the result is the integer -2 or 2. QED