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Consider the meromorphic functionClose the contour with a large semicircle in the upper half-plane, indenting above the three real pole $-1,0,1$. The large-arc integral vanishes by Jordan's lemma, while the upper-half-plane indentation rule givesThe three residue areTheir sum is $\cos 1-1$, so the residue theorem yields

The real part of the integrand is an odd function, so its full-line principal value is zero. Its imaginary part,is an even function. Taking imaginary parts and halving the full-line integral therefore gives

Use the upper half-disc of radius $R$, with a clockwise semicircular indentation of radius $\epsilon$ above zero. The integrand is holomorphic inside that indented contour, so its total integral is zero. On the large arc,because $a+b>0$, $a-b>0$ and $\mathrm{Im}z\ge 0$. Its integral has modulus at most $\pi /R$ and vanishes as $R\to \infty$.

On the small clockwise arc, $f(z)=b/z+O(1)$, so the integral tends to $-i\pi b$. Therefore the remaining real-axis integral satisfiesThe imaginary part is even and is an ordinary absolutely convergent integral: near zero its integrand tends to $ab$, and at infinity it is $O(x^{-2})$. Taking imaginary parts givesThe complex integral needs the Cauchy principal value because its real part has a $b/x$ singularity; the requested sine-product integral does not. This sine-product integral with inverse-square weight is an application of the upper-half-plane indentation rule.

By symmetry assume $a\ge b>0$ and integrate $f(z)=e^{iaz}\sin (bz)/z^2$ around an upper semicircle indented clockwise above zero. The residue at zero is $b$. The expansion $(e^{i(a+b)z}-e^{i(a-b)z})/(2i{z}^2)$ bounds the large arc by $\pi /R$, including when $a=b$. The upper-half-plane indentation rule therefore gives $\mathrm{PV}{\int }_{\mathbb{R}}f(x)dx=i\pi b$. Its imaginary part is even and ordinarily integrable, yielding the displayed value. At zero the sine-product integrand tends to $ab$; at infinity it is bounded by $1/x^2$. The principal value is needed for the auxiliary complex integral, not for the real sine-product integral.