Answer
$\int_Cf(x) dx+g(y) dy=0$
Work Step by Step
We need to show that $\int_Cf(x) dx+g(y) dy=0$
Here, $f$ and $g$ are differentiable functions and $C$ is any piece-wise smooth simple closed plane curve.
Green's Theorem states that $\int_C Adx+B dy=\iint_D (\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y})dA$
Here, $D$ is the region enclosed inside the counter-clockwise oriented loop $C$.
Here, we have $\int_Cf(x) dx+g(y) dy=\iint_D (\dfrac{\partial g(y)}{\partial x}-\dfrac{\partial f(x)}{\partial y})dA$
Thus, $\int_Cf(x) dx+g(y) dy=0$
Hence, the result has been proved.
