Answer
$$0$$
Work Step by Step
We are given that $F(x,y, z)=(y-z, z-x, x-y)$
Also, the curve is $r(t)= (\cos t, \sin t, \cos t)$
This implies that $r'(t) = (-\sin t, \cos t, -\sin t)$
Therefore, the integral is:
$\oint F \ dr=\int_0^{2 \pi} F[r(t)] r'(t) \ dt\\=\int_0^{2 \pi} (\cos t, \sin t, \cos t) (- \sin t, \cos t, -\sin t)\\=\int_0^{2 \pi} 0 \ dt \\=0 $
