I wish to understand how to take complex integrals and derivatives.
For the simplest case, $\int{x+iy, dz}$
$f(z) dz = (u + iv)(dx + idy) = (u + iv) dx + (iu − v) dy = (udx − vdy) + i(vdx + udy)$
So $\int({x+iy})(dx+idy)$ $=(\int{xdx}-\int ydy)+i(\int ydx+\int xdy)$ $=\frac{x^2}{2}-\frac{y^2}{2}+2xyi$
Did I do it right?
Almost, you have $2xyi$ instead of $xyi$. You can also see that as $$\int z\,{\rm d}z = \frac{z^2}{2},$$since $$\frac{(x+iy)^2}{2} = \frac{x^2-y^2}{2} + xyi.$$