(i) Upper half of |z| = 2
|z| = 2; z = 2e$^{i \theta}$
dz = 2i$^{i \theta} \,\, d \theta $
$
\int_c \frac{2z + 3}{2} \,\, dz \\
= \int_0^{\pi} \frac{2(2^{i \theta})}{2^{i \theta}} 2i^{i \theta} \,\, d \theta \\
= i \int_0^{\pi} (4e^{i \theta} + 3) \,\, d\theta \\
= i [\frac{4 e^{i \theta}}{i} + 3 \theta]_0^{\pi} \\
= i [\frac{4e^{i \pi}}{i} - \frac{4e^0}{i} + 3\pi] \\
= i [\frac{-4}{i} - \frac{4}{i} + 3 \pi] \\
= i[\frac{-8}{i} + 3 \pi] \\
= 3 \pi i - 8
$
(ii) Lower half of |z| = 2
$
|z| = 2; \\
z = 2e^{i \theta}; \\
dz = 2ie^{i \theta} \,\, d \theta \\
\int_{\pi}^{2\pi} \frac{2(2e^{i \theta} + 3)}{2e^{i \theta}}(2ie^{i \theta}) \,\, d \theta \\
i \int_{\pi}^{2\pi}(4e^{i \theta} + 3) \,\, d\theta \\
= i[\frac{4e^{i \theta}}{i} +3 \theta]_{\pi}^{2\pi} \\
= i (\frac{8+3\pi i}{i}) \\
= 3 \pi i + 8
$
(iii) Whole circle in anti-clockwise direction
$ \theta = 0 \to 2\pi $
$
z = 2e^{i \theta} \\
z = 2ie^{i \theta} \,\, d\theta \\
\int_{0}^{2\pi} \frac{2(2e^{i \theta} + 3)}{2e^{i \theta}}(2ie^{i \theta}) \,\, d \theta \\
= i \int_{0}^{2\pi}(4e^{i \theta} + 3) \,\, d\theta \\
= i[\frac{4e^{i \theta}}{i} +3 \theta]_{0}^{2\pi} \\
= i [\frac{4(1)}{i} + 6 \pi - \frac{4}{i}] \\
= 6 \pi i
$