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I need to compute the definite integral defined as $$\int_{-\infty }^{+\infty } \frac{\sin ({x_0}\, \omega )-\sin (x \omega )}{(\sin (x \omega )-\sin ({x_0}\, \omega ))^2+(x-{x_0})^2} \, dx\,.$$

When I try

Integrate[(- Sin[x \[Omega]] + 
    Sin[x0 \[Omega]])/((x - x0)^2 + (Sin[x \[Omega]] - 
     Sin[x0 \[Omega]])^2), {x, -\[Infinity], +\[Infinity]}] // \
Simplify

Mathematica cannot find the result.

So could you help me? Thank you.

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  • $\begingroup$ Warning: This user does not mark any answers as accepted. Proceed with posting answers to this user's question with due vigilance. $\endgroup$ – QuantumDot Oct 11 at 20:09
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A numerical solution, considering the singularity at x==x0, might be

int[x0_?NumericQ, \[Omega]_?NumericQ] := NIntegrate[(-Sin[x \[Omega]] + Sin[x0 \[Omega]])/((x - x0)^2 + (Sin[x \[Omega]] - Sin[x0 \[Omega]])^2), {x, -\[Infinity], +\[Infinity]},, Method -> "PrincipalValue", Exclusions -> {x == x0}]

int[1,1]
(*-0.865516*)
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