The localization $K[X]\_{X}$ is a ring extension of $K[X].$ I want to show that $K[X]\_X$ is not integral over $K[X]$ using lying above.

I tried to find a maximal ideal in $K[X]\_X$ whose contraction in $K[X]$ is not maximal ideal, or maybe we can produce two prime ideals above one containing another whose contraction is the same prime ideal. I need some help to construct such prime ideals. Thanks.