$\displaystyle \int_C (z*^2dz + z^2dz*)$, where C is the curve defined by $\displaystyle z^2+2|z|^2+z*^2=2(1-i)z+2(1+i)z*$
$\displaystyle z$ is a complex number x+iy, $\displaystyle z*$ is his complex conjugate
Thanks
I think that there is a problem with your curve.
first I will use $\displaystyle z*=\bar{z}$ for complex conjugation
recall that
$\displaystyle |z|^2=z\bar{z}$ so we can factor the left hand side as follows
$\displaystyle z^2+z\bar{z}+z\bar{z}+\bar{z}^2=z(z+\bar{z})+\bar{ z}(z+\bar{z})=(z+\bar{z})^2=(2\text{Re}(z))^2=4x^2$
Now for the right hand side we get
$\displaystyle 2(1-i)(x+iy)+2(1+i)(x-iy)=2[(x+iy-ix+y)+(x-iy+ix+y)]=4x+4y$
This gives the curve as
$\displaystyle 4x^2=4x+4y \iff y=x^2-x$
If you expand out the integrand in a similar fashion you will get
$\displaystyle (2x^2-2y^2)dx+4xydy$
remember that
$\displaystyle dz=dx+idy; \quad d\bar{z}=dx-idy$
So this gives the real line integral
$\displaystyle \oint (2x^2-2y^2)dx+4xydy$
but the curve of integration (the parabola) is unbounded so the integral will be as well.