This problem is from Guillemin and Pollacks "Differential Topology" (p. 192, 2):
Let $\gamma$ be a closed curve in $\mathbb{R}^2-\{0\}$ and $w$ any closed $1$-form on $\mathbb{R}^2-\{0\}$. Prove that \begin{equation} \oint_{\gamma} w=W(\gamma,0)\int_{S^1} w, \end{equation} where $W(\gamma,0):=\text{deg}(\gamma/|\gamma|)$ is the winding number of $\gamma$ with respect to the origin.
In particular conclude that \begin{equation} W(\gamma,0) = \frac{1}{2\pi }\oint_{\gamma} d\text{ arg}. \end{equation}
I am quite sure that one must use the degree formular proven in the previous section given by \begin{equation} \int_X f^*w = \text{deg}(f)\int_Y w \end{equation} for $f:X\rightarrow Y$ a smooth map, $Y$ connected, $X$ and $Y$ compact and oriented.
However, the task seems strange to me because in the book the definition of the circular integral is given by \begin{equation} \oint_{\gamma} w := \int_{S^1}\gamma^* w \end{equation} where $\gamma:S^1\rightarrow Z$ for some manifold $Z$. So if $w$ is a 1-form on $Z=\mathbb{R}^2-\{0\}$, then the last expression, $\int_{S^1}w$, is not even well defined. Is there an error in the task and the result should actually be $W(\gamma,0)\int_{S^1}h^*w$ where $h$ is some parameterisation of $w$ around $S^1$? And if so, I am not sure how to create an expression of the form $\int_{S^1}(\gamma/|\gamma|)^*h^*w$ to use the degree formular. Thanks a lot.
PS: I know that this is a possible double but there are many errors in the OP's question and the answer below only corrects some of them and does not provide a satisfactory/complete solution. In particular the well-definedness of the last expression is not discussed.