I recently started learning some elementary stuff about complex functions, one thing I came across was the integral of $z^{n}$ around a circle of radius $r>0$ in counterclockwise direction. For $n\neq -1$, this is 0, and for $n=-1$, this is $2πi$ which can be seen from Cauchy's integral formula. It is possible to split this integral into complex sum of real integrals in this manner: $$z^{-1} = \frac{x}{x^2+y^2} - i \frac{y}{x^2+y^2}$$ Then, $$I = \int_{C'} \frac{x}{x^2+y^2}dx + \frac{y}{x^2+y^2} dy + i \int_{C'} \frac{x}{x^2+y^2}dy - \frac{y}{x^2+y^2} dx$$ The real part is 0, for the imaginary part, I use $x = \sqrt{r^2-y^2}, dx = \frac{-y}{x} dy$ (I use positive part of circle because as y goes from -r to r, we get the right half of circle ($x>0$) for counterclockwise orientation). This then gives: $$i \int_{-r}^{r} \frac{\sqrt{r^2-y^2}}{r^2}dy - \frac{y}{r^2} \frac{-y}{\sqrt{r^2-y^2}}dy $$
Which simplifies to $$I = i \times \left. \arcsin\left(\frac{y}{r}\right)\right|_{-r}^{r} = iπ$$
Where did I go wrong? This has been bothering me for quite a while now. Thanks for reading.