I'm reviewing complex analysis for the upcoming GRE. The following question is Problem 4.42 from Schaum's Outline of Complex Variables.
Evaluate $\int_C \overline{z}^2 dz + z^2 d\overline{z}$ along the curve $C$ defined by $z^2 + 2z\overline{z} + \overline{z}^2 = (2-2i)z + (2+2i)\overline{z}$ from the point $z = 1$ to $z = 2+2i$.
The constraint simplifies to $y= x^2 -x$, and using the substitution $dz = dx+ i\ dy$ and $d\overline{z}=dx-i\ dy$, the integrand can also be simplified to: $$ \int_C x^2 dx + (2xy +iy^2) dy $$
Of course, I can just plug in $y=x^2 - x$ and integrate the resultant polynomial equation term by term, but is there an easier way to do this question? I think there might be some symmetry that I can exploit, judging by the funky nature of $\overline{z}^2 dz + z^2 d\overline{z}$, but I'm not familiar enough with complex analysis to see what it might be.