My question is regarding the following complex integral:
$$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$
where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq \frac{\pi}{2}$.
Below is my solution:
\begin{equation} \begin{split} \int_\gamma\frac{\operatorname{Arg}(z)}{z} dz =& \int_0^\frac{\pi}{2}\frac{t}{e^{it}}*ie^{it}dt \\ = & \; i\int_0^\frac{\pi}{2}tdt \\ =& \;\frac{\pi^2i}{8} \end{split} \end{equation}
I believe my attempt at this question is correct but the allowable multiple choice answers are:
a. $0$ $\quad$ b. $-\frac{i\pi}{2}$ $\quad$ c. $\frac{i\pi}{2}$ $\quad$ d. $\frac{\pi}{2}$ $\quad$ e. $\ln\left(\frac{\pi}{2}\right)$
Is there an error in my work or maybe just a typo in the MC answers?
Thanks in advance for you help!