Corollary 2
Tag 04O5, permanent. Cite it as [Bourbaki, Tag 04O5].
Commutative Algebra, chapter V, § 1. Notion of an integral element
With the same hypotheses and notation as in Corollary 1, Tr_{K'/K}(x) and N_{K'/K}(x) are integral over A.
Tr_{K'/K}(x) and N_{K'/K}(x) are, to within a sign, coefficientsof Pc_{K'/K}(x; X) (Algebra, Chapter VIII, § 12, no. 1, equations (4))and hence are integral.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.