Solve Improper Integrate using Residue theorem for this integral [migrated]


I am trying to solve this integral

$ \int_{-\infty}^\infty \frac{x\cdot sin(x)}{x^2+4} \cdot dx $

I have applied the residue theorem on a semicircle of radius $ R> 2$ , $ \gamma$ , so I have

$ \int_\gamma \frac{z\cdot sin(z)}{z^2+4} \cdot dz = \int_{-R}^R \frac{x\cdot sin(x)}{x^2+4} \cdot dx+ \int_{C_R} \frac{z\cdot sin(z)}{z^2+4} \cdot dz$

where $ C_R = \{ z : z=R e^{i \theta} , \theta \in (0,\pi)\}$ , but I cannot limit the second integral to eliminate this contribution when $ R \to \infty$ and I don’t know how I could do the integral in another way