By Green's theorem,
[tex]\displaystyle\oint_C(5y^3\,\mathrm dx\pm5x^3\,\mathrm dy)=\iint_D\left(\pm\frac\partial{\partial x}[5x^3]-\frac\partial{\partial y}[5y^3]\right)\,\mathrm dx\,\mathrm dy[/tex]
where [tex]D[/tex] is the disk of radius 2 centered at the origin.
Differentiating and converting to polar coordinates, you have
[tex]\displaystyle\iint_D\left(\pm15x^2-15y^2\right)\,\mathrm dx\,\mathrm dy=15\int_0^{2\pi}\int_0^2\left(\pm r^2\cos^2\theta-r^2\sin^2\theta\right)r\,\mathrm dr\,\mathrm d\theta=15\int_0^{2\pi}\int_0^2\left(\pm \cos^2\theta-\sin^2\theta\right)r^3\,\mathrm dr\,\mathrm d\theta[/tex]
which, depending on the sign concealed by the question mark, could be reduced to either
[tex]\displaystyle15\int_0^{2\pi}\int_0^2r^3\cos2\theta\,\mathrm dr\,\mathrm d\theta=0[/tex]
if a plus, or
[tex]\displaystyle-15\int_0^{2\pi}\int_0^2r^3\,\mathrm dr\,\mathrm d\theta=-8\pi[/tex]
