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Calculus Level 4

True or false:

Let \( \phi: [0, 2\pi ] \rightarrow \mathbb R^2 , \quad \phi(t) = (2+ \cos t , \sin t ) \) and let \( \gamma: [0, 2\pi ] \rightarrow \mathbb R^2 , \quad \gamma(t) = ( \cos t , \sin t ) \), then the equation below is satisfied.

\[ \int_{\phi} \dfrac{-y}{x^2+ y^2} \, dx + \dfrac x{x^2+y^2} \, dy = \int_{\gamma} \dfrac{-y}{x^2+ y^2} \, dx + \dfrac x{x^2+y^2} \, dy \]

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