Problem

(R) Let $C$ be the counterclockwise oriented boundary of the region in the $xy$-plane enclosed by $x^2+y^2-2x=0$ and $x^2+y^2-2y=0$. Evaluate the line integral $\oint_C (y+e^{-x^2})\,\mathrm{d}x+(3x+\sin(y^2))\,\mathrm{d}y$. Answer: $\underline{\quad(20)\quad}$.