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 $\mathrm{char}\,F = p>0$, the only one is $R=-\mathrm{id}$. When $\mathrm{char}\,F = 0$, we have either $R = -\mathrm{id}$ 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 $F[x]$.