This is so because your ellipse is simply a stretching of a unit circle (x = cos t, y = sin t) along the axes. No rotation is involved.
Now we transform the integral to the variable , using ellipse parametric equations:
Find the lower integral limit:
Find the upper integral limit [we take the value x = 9, because the maximum value of cosine is a unit]:
So we have
Finally we have
See this picture