## Problem on a line integral over a circular arc

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 $(0,1)$.
• ## Solution

We recall:
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.
Recall that:
We recall the fundamental theorem of line integrals:
Before we can apply the theorem, we need to find $\phi$.
Applying the fundamental theorem of line integrals: \begin{align}\int_C \mathbf{F}(x,y)\cdot d\mathbf{r} &= \phi(0,1) - \phi(1,0) \\ &= \log(1) - \log(1)= 0.
\end{align}
We conclude $$\int_C \left( \frac{1}{x+y} \mathbf{i} + \frac{1}{x+y} \mathbf{j} \right) \cdot d\mathbf{r} = 0.$$