(a) Let $γ_1 = [1, 0] ∪ [0, i]$. Compute $\int_{γ1} \bar{z} dz$.
(b) Let $C$ be the circle of radius $1$ and center $0$, and let $γ_2$ be the (smaller) arc of $C$ joining the points $1$ and $i$. Compute $\int_{γ_2} \bar{z} dz$.
(c) Only based on (a), (b), can we tell that $\bar{z} $ does not admit a holomorphic primitive on $\mathbb C$?
My attempts:
a) Let $z = x+iy$, then $\bar{z} = x-iy$ and so $$ \int_{γ1} \bar{z} dz = \int_1^0 x dz + i \int_1^0 - y dz + \int_0^i x dz + i \int_0^i - y dz = x+y + i(x-y) $$
b) $$\int_{γ_2} \bar{z} dz = \int_0^1 x dz + i \int_0^1 - y dz = x-iy$$
c) I'm not sure about it, but I think that yes, we can tell that it doesn't admit a holomorphic primitive on $\mathbb C$ since the integral in part a) and b) are not equal.
Your answers to (a) and (b) don't make sense, since an integral is a number.
Concerning (a), we have\begin{align}\int_{\gamma_1}\overline z\,\mathrm dz&=\int_0^1-t\,\mathrm dt+\int_0^1(it)\times i\,\mathrm dt\\&=-1.\end{align}And, concerning (b), we have\begin{align}\int_{\gamma_2}\overline z\,\mathrm dz&=\int_0^{\pi/2}e^{-it}\times ie^{it}\,\mathrm dt\\&=\frac{i\pi}2.\end{align}
Since this gives you distinct numbers, the conjugation does not admit a holomorphic primitive.