Find $\int_C \bigl( \frac{1}{x+y} \mathbf{i} + \frac{1}{x+y} \mathbf{j} \bigr) \cdot d\mathbf{r}$, where $C$ is the segment of the unit circle going counterclockwise from $(1,0)$ to $(0,1)$.
We begin by trying to apply the fundamental theorem to $\int_C \mathbf{F}(x,y) \cdot d\mathbf{r}$, where $\mathbf{F}(x,y) = \frac{1}{x+y} \mathbf{i} + \frac{1}{x+y} \mathbf{j}$. If this approach does not work, we will try one of the others.