Let $\alpha$ be the closed curve along the square with vertices at $1, i, -1, -i$.
Give an explicit parametrization for $\alpha$ and calculate $$\frac{1}{2\pi i}\int_\alpha\frac{dz}{z}$$
I got this parametrizations: $$\alpha(t) = \begin{cases} 1+(i-1)t & 0\leq t\leq 1\\ i+(-1-i)(t-1) & 1\leq t\leq 2\\ -1+(-i+1)(t-2) & 2\leq t\leq 3\\ -i+(1+i)(t-3) & t\leq 3\leq 4\end{cases}$$ So far so good, but now I want to find the asked integral. I want to compute the four integrals apart. For the first part of the path we thus have $\int_0^1\frac{i-1}{1+(i-1)t}dt$. But how can I calculate a primitive?
Hint
Substitute $u = \frac t{i-1}$ and see what happens.