How to evaluate this complex integral?

Question

I want to evaluate this integral

$$\displaystyle\oint \dfrac{x\,dx+(y-1)\,dy}{x^2+(y-1)^2}$$

using this $$\int \frac{1}{z-1}\,dz$$

where $z\in C, C:z(\theta)=2.e^{i\theta},0 \le \theta \le 2\pi$

My try : i tried to express the denominator in this form $|z-i|^2=x^2+(y-1)^2$ and $d(z-i)=dx+idy$ what to do after this ? this is where i got stuck

Answer 1

$\DeclareMathOperator{\Re}{Re}$Presumably you're intended to write $z = x + iy$, so that $$ \Re \frac{dz}{z - i} = \Re \frac{(\overline{z - i})\, dz}{|z - i|^{2}} = \Re \frac{(x - (y - 1)i)\, (dx + i\, dy)}{x^{2} + (y - 1)^{2}} = \frac{x\, dx + (y - 1)\, dy}{x^{2} + (y - 1)^{2}}. $$