I have a problem asking me to find $\int_C \textbf{f} \cdot d\textbf{r}$ where $\textbf{f}$ = $(\sin y,x\cos y)$, and the curve $C$ is any closed circle.
I'm struggling with this, so far I have found that $d\textbf{r}$ is $(-rsin\theta,rcos\theta)d\theta$, which is then $(-y,x)d\theta$ in cartesian coordinates.
I then tried to do the following; $\int_0^{2\pi} (\sin y,x\cos y) \cdot (-y,x)d\theta$, but I'm not making any progress and I'm unsure whether my method is correct?
You're not applying the formula for line integrals correctly. Remember that it's $$\int_C \mathbf f\cdot d\mathbf r = \int_{t_0}^{t_1} \mathbf f(\mathbf x(t))\cdot \mathbf x'(t)\ dt$$
where $C$ is parametrized by $\mathbf x(t)$ over $t\in [t_0, t_1]$. Notice that the only variable is $t$ (you can't mix in three different variables like you do in your proposed integral).
However, that's not the way to do this problem. The most reasonable parametrization of a circle leads to an integral that is very difficult to solve. Instead:
Hint: What do you know about the value of a line integral $\int_C \mathbf f\cdot d\mathbf r$ over a closed curve $C$ when $\mathbf f$ is a conservative field? Is there a way to check whether this is a conservative field?