# Computing the definite integral of a fractional polynomial containing sin(x) and x^n

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.

• Warning: This user does not mark any answers as accepted. Proceed with posting answers to this user's question with due vigilance. – QuantumDot Oct 11 '19 at 20:09
• I am sorry for that because I don't know "flag or mak it as accepted answer". Now I understand how it works. – tanghe2014 May 2 '20 at 0:16

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}]