Answer:
<h2>WOW!!What an amazing question</h2>
Direct computation:
Parameterize the top part of the circle by
with , and the line segment by
with . Then
Using the fundamental theorem of calculus:
The integral can be written as
If there happens to be a scalar function such that , then is conservative and the integral is path-independent, so we only need to worry about the value of at the path's endpoints.
This requires
So we have
which means is indeed conservative. By the fundamental theorem, we have