Evaluate $ \ \int_{- \infty}^{\infty} \frac{x \sin (3x)}{x^2 +4}\,dx \ $ using Jordan's Lemma

Question

How to to evaluate $ \ \int_{- \infty}^{\infty} \frac{x \sin (3x)}{x^2 +4}\,dx \ $ using Jordan's Lemma?

Answer 1

Hint: Consider the complex integral

$$ \int_{C} \frac{ z\,{\rm e}^{3 i z} }{ (z^2 + 4)} dz \,.$$

See here for the contour $C$.