For example, let $C$ be the unit circle centered around the origin oriented clockwise, and let $$\vec F = \frac{-y\vec i+ x\vec j}{x^2+y^2}$$
Then (I think) $F$ is tangent to $C$ at every point. So is there an easy way to calculate the line integral $\int_C\vec F\cdot d\vec r$ ? One guess is that the line integral is just the length of $C$ multiplied by the magnitude of $F$, but I have no idea why (I don't know how to formally justify that, if it's even correct).
Use the usual parametrization of the unit circle $\;r(t)=(\cos t,\,\sin t)\;,\;\;t\in[0,2\pi]\;$, so
$$\oint_C\frac{-y\,dx+x\,dy}{x^2+y^2}=\int_0^{2\pi}\left((-\sin t,\,\cos t\right)\bullet\left((-\sin t,\,\cos t)\right)dt=\int_0^{2\pi}dt=2\pi$$