Find the value of the integral $\oint_C (x^3+x)dx+(1+y^2+y^3)dy$ , where $C=\{(x(t),y(t))| x(t)=2+3\cos t ,y(t)=5+4\sin t , 0\leq t<2\pi\}$
My work :
I think the answer is $0$
$C: (\frac{x-2}{3})^2+(\frac{y-5}{4})^2=1$ , an eclipse . Note that $(x^3+x)dx+(1+y^2+y^3)dy=d(\frac{x^4}{4}+\frac{x^2}{2}+y+\frac{y^3}{3}+\frac{y^4}{4})$ .
So the line integral is integrated over a curve where initial point and end point coincide and the integrand is an exact differential. So the line line integral$=0$