For context I am working on Atiyah-Macdonald 5.4.
I want to show that the extension $k[x^2-1]\subset k[x]$ is integral. I believe this is the case using that $k[x^2-1]=k[x^2]$, which I believe can be proved using
$$(x^2-1)+1=x^2$$
This equality feels, and looks, obviously true. But if it is, why present the ring as $k[x^2-1]$ and not $k[x^2]$?
I am either missing something obvious, or there is some "pedagogical" reason which isn't clear. Any help is appreciated.