Answer
The integral is $0$ irrespective of the curve $C$.
Work Step by Step
The tangential form for Green's Theorem is given as:
Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy$
$\implies \oint_C4x^3 \ dx +x^4 \ dy= \iint_{R} (\dfrac{\partial (x^4) }{\partial x}-\dfrac{\partial (4x^3 y) }{\partial y}) dx dy= \iint_{R} [ 4x^3-4x^3 ] \ dx \ dy$
and, $\oint_C4x^3 \ dx +x^4 \ dy=0$
This means that the integral is $0$, irrespective of the curve $C$.
