 Solution

The contour integral along a piecewise smooth path is defined by parametrization:with a sum over the smooth pieces. The residue theorem states that for a positively oriented simple closed contour and a function holomorphic on and inside it apart from finitely many isolated singularities $a_j$ inside,More generally a closed contour uses the winding number of each singularity as its weight.

Set $z=e^{it}$, so $dt=dz/(iz)$ and $\cos t=(z+z^{-1})/2$. The required real integral becomesThe denominator factors as $(2z^2+1)(z^2+2)$. Only the simple poles $z=\pm i/\sqrt{2}$ lie inside the unit circle. At either one,The residue theorem now gives