Answer
$\int_Cf(x) dx+g(y) dy=0$
Work Step by Step
prove that $\int_Cf(x) dx+g(y) dy=0$
where $f$ and $g$ are differentiable functions and $C$ is any piece-wise -smooth simple closed plane curve.
Consider the Green's Theorem: $\int_C fdx+g dy=\iint_D (\dfrac{\partial g}{\partial x}-\dfrac{\partial f}{\partial y})dA$
Here, $D$ is the region enclosed inside the counter-clockwise oriented loop $C$.
$\int_Cf(x) dx+g(y) dy=\iint_D (\dfrac{\partial g(y)}{\partial x}-\dfrac{\partial f(x)}{\partial y})dA$
Thus, $\int_Cf(x) dx+g(y) dy=\iint_D (0-0)=0$
Hence, the result is verified.
