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Consider the line integral

$$\int_{c} (xdy-ydx)$$

the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ rectangle and a semi-circle of radius $1$. The line integral evaluates to

  1. $6+ \dfrac{\pi}{2}$
  2. $8+\pi$
  3. $12+\pi$
  4. $16+2\pi$
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