Since $r^n+a_1r^{n-1}+...+a_n=0$ is an equation of integral dependence of $r$ over an ideal $I$, by definition, $a_i\in I^i$. So $\displaystyle a_i=\sum_{k=1}^{k=n(i)}a_{k1}^{(i)}\ldots a_{ki}^{(i)}$, with $a_{kj}^{(i)}\in I$. Take for $J$ the ideal generated by the $a_{kj}^{(i)}$.
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