Why is the projective space an integral scheme

Question

I would like to prove that the $Proj(k[x_{0},\dots, x_{n}])$ is an integral scheme ($k$ is a field). I know a preposition that says a scheme is integral if and only if the image of the structure sheaf on every open is an integral ring. Of course I know that the image of basic open subsets is an integral ring, but can I derive that for every open? Or is there any other way of proving this?