= Solution
Put $z=e^{i\theta}$, so $d\theta=dz/(iz)$, $\cos\theta=(z+z^{-1})/2$ and $\sin\theta=(z-z^{-1})/(2i)$. Rational expressions in these <Trigonometric function>[trigonometric functions] become rational <contour> integrands on the <unit circle>. The method is especially useful when the resulting denominator has a small number of identifiable <Pole>[poles]. A <pole> on the <integration> circle requires a separately specified <principal value> or indentation; the <residue theorem> does not automatically assign an ordinary <integral> there.
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