Holomorphical extension to the Annulus

Question

Let $D=\{z:1<|z|<2\}$ and $f$ is holomorphic on $D$. Suppose that f has a primitive $f_1$ on D and $f_1$ also has a primitive $f_2$, etc for every $n$ $f_n$ has a primitive $f_{n+1}$ in $D$. How can we show that $f$ extends holomorphically to $D$? I know that $f$ has a primitive on $D$ if and only if $\int_{|z|=1}f(z)dz=0$. But I couldn't proceed it further. Please any help.