= Solution
As an $A'$-algebra, $A\otimes_RA'$ is generated by the elements $a\otimes1$. If $a$ obeys a monic relation
$$
a^d+r_{d-1}a^{d-1}+\cdots+r_0=0
$$
over $R$, then $a\otimes1$ obeys the same monic relation after applying the structure map $R\to A'$. Thus every generator is an <integral element>. The <subalgebra generated by finitely many integral elements> is finite as a module, and therefore integral; each tensor involves only finitely many generators. Hence $A\otimes_RA'$ is integral over $A'$.
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