Chapter 15.4, Problem 39ES

Evaluate the integral ${\displaystyle {\int }_C\text{F}\cdot d\text{r},}$ where C is the boundary of the region R and C is oriented so that the region is on the left when the boundary is traversed in the direction of its orientation.

$\text{F}\left(x,y\right)=\left(x^2+y\right)\text{i}+\left(4x-\cos y\right)\text{j};C$ is the boundary of the region R that is inside the square with vertices $\left(0,0\right),\left(5,0\right),\left(5,5\right),\left(0,5\right)$ but is outside the rectangle with vertices $\left(1,1\right),\left(3,1\right),\left(3,2\right),\left(1,2\right).$