When is $F[X]/(f)$ over $F$ an integral extension?

Question

I was looking at the concept of integral extension. So $K/F$ is integral means that for every $a \in K$ is the root of a monic polynomial with coefficient in $F$. For example $\mathbb{Q}/\mathbb{Z}$ is not integral since $1/3$ is not the root of a monic polynomial with integral coefficients. I was wondering when the extension $F[X]/(f)$ over $F$ was an integral extension. I was suggested it is integral if and only if $f$ is a monic polynomial? Why? If I take for instance $[p]_f \in F[X]/(f)$ it is not a root of $f$ I think.