I am struggling with the following problem. Let $S$ be a Integral domain and $K$ its Quotient field. Furthermore let $A/B/K$ be Field extensions. We set $X = \text{Int}_S(B)$ where $\text{Int}_S(B)$ is the integral closure. Then it follows that $$\text{Int}_S(A) = \text{Int}_X(A)$$
So all i need to show is that $X = \text{Int}_S(B) = S$. I know that for a Integral domain $S$ and for its Quotient field $K$ it follows that $\text{int}_S(K) = S$ but i am not sure on how to use this fact here.
If $C/D$ is integral and $D/E$ is integral then $C/E$ is integral, this is because we define integral as the elements $a$ such that $E[a]$ is contained in a finitely generated $E$-module
(the theorem is that the finitely generated condition is equivalent to being a root of a monic (non-irreducible) polynomial $\in E[x]$).
Thus $Int_{Int_S(B)}(A)\subset Int_S(A)$.
The other direction $Int_S(A)\subset Int_{Int_S(B)}(A)$ is obvious from $S\subset Int_S(B)$.