Injective Rota-Baxter operators of weight 1 on $F[x]$

We describe all injective Rota-Baxter operators $R$ of weight 1 on the polynomial algebra $F[x]$. When $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}F=p>0$, the only one is $R=-\mathrm{i}\mathrm{d}$. When $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}F=0$, we have either $R=-\mathrm{i}\mathrm{d}$ or, up to conjugation with automorphisms of $F[x]$, $R(1)=x$, and $R$ is uniquely defined via this equality. Together with the known weight-zero case, this completes the classification of injective Rota-Baxter operators of any weight on $F[x]$.