Calculate $\displaystyle\int_C \frac {x^3dy-y^3dx}{x^{5/3}+y^{5/3}}$ where $C:x^{2/3}+y^{2/3}=a^{2/3}$ in the first quadrant.
My attempt:
Parametrization: $$x(t)=a\cos^3t$$ $$y(t)=a\sin^3t$$
Then the integral becomes $$I=\int_0^{\frac {\pi}2}\frac {\sin^2t\cos^{10}t+\cos^2t\sin^{10}t}{\cos^5t+\sin^5t}dt$$
And I'm stuck here.