Antiderivative of infinitely often differentiable function

Define interval $I=(0,1)$, ${C}^{\mathrm{\infty}}(I)$ as a ring of infinitely often differentiable functions from I to $\mathbb{R}$, and ${C}_{C}^{\mathrm{\infty}}(I)\subset {C}^{\mathrm{\infty}}(I)$ an ideal of functions having compact support. Set $d:{C}^{\mathrm{\infty}}(I)\to {C}^{\mathrm{\infty}}(I)$ and $\overline{d}:{C}_{C}^{\mathrm{\infty}}(I)\to {C}_{C}^{\mathrm{\infty}}(I)$ a usual differentiation.

Is it true that $\text{Im}(d)={C}^{\mathrm{\infty}}(I)$ and $\text{Im}(\overline{d})={C}_{C}^{\mathrm{\infty}}(I)$? In other words, is it true that antiderivative F of $f\in {C}^{\mathrm{\infty}}(I)$ also lies in ${C}^{\mathrm{\infty}}(I)$ and same for ${C}_{C}^{\mathrm{\infty}}(I)$? If not, how can one describe $\text{Im}(d)$ and $\text{Im}(\overline{d})$?