So I'm practicing a few problems and I can't get this one -
$P(x,y) = e^x \sin(y) \\ Q(x,y) = e^x \cos(y)$
$C$ is the right hand loop of the graph of the polar equation $r^2 = 4\cos(\theta)$
I want to evaluate:
$\int_{C}{P(x,y)\:dx+Q(x,y)\:dy}$
Now I tried the right hand side of Green's theorem, but it's difficult because $\frac{dp}{dy}$ in polar has a $\cos(r\sin(\theta))$ term in it.
If I just try parametrizing with $x = a\sin(t)$ and $y = a\cos(t)$, where $-\pi/2 \leq t \leq -\pi/2$, then I get another ludicrous integral with $e^{a\cos(t)}\sin(a\sin(t))a\cos(t)\:dt$ as $P \:dx$, which seems insane to solve.
So I think I may be missing some trick in doing this problem. What am I missing and how should I do this?