EXAMPLE $2$ Evaluate $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}(2y-e^{sin(x)})dx(6x\sqrt[\phantom{2}]{y^{4}1})dy,$ where $C$ is the circle $x^2{y}^2=25.$

SOLUTION The region $D$ bounded by $C$ is the disk $x^2{y}^{2}25,$ so let's change to polar coordinates after

$[$: $(1,2y-e^{sin(x)})\}$

$={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$

$=4{\displaystyle \underset{0}{\overset{2}{}}}d{\displaystyle \underset{0}{\overset{5}{}}}rdr$

$=$