# 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