Answer
$\int_Cf(x) dx+g(y) dy=0$
Work Step by Step
Apply Green's Theorem as: $\int_C adx+b dy=\iint_D (\dfrac{\partial b}{\partial x}-\dfrac{\partial a}{\partial y})dA$
where, $D$ shows the region enclosed inside the counter-clockwise oriented loop $C$.
Our aim is to prove 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.
Therefore,
$\int_Cf(x) dx+g(y) dy=\iint_D [\dfrac{\partial g(y)}{\partial x}-\dfrac{\partial f(x)}{\partial y}]dA$
This implies that $\int_Cf(x) dx+g(y) dy=0$
Hence, the result is verified.
