= Solution
The <Krull intersection theorem> gives
$$
\bigcap_{r\geq0}(x^r)=0
$$
because $(R,\mathfrak m)$ is a <Noetherian local ring> and $x\in\mathfrak m$. Consequently every nonzero $a\in R$ has a largest $x$-adic order: one can write
$$
a=x^ra',\qquad a'\notin(x).
$$
Suppose nonzero elements $a,b$ satisfy $ab=0$. Write $a=x^ra'$ and $b=x^sb'$ with $a',b'\notin(x)$. Since $x$ is a <non-zero-divisor>, cancellation of $x^{r+s}$ gives $a'b'=0$. Reducing modulo $(x)$ now gives a product of two nonzero elements equal to zero in $R/(x)$, contradicting that this quotient is an <integral domain>. Hence $R$ is an integral domain.
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