10.7.2 Calculating integrals of a real variable using residues

Problem: Calculate the integral, using residues: \[ I_{2}=\int_{-\infty}^{+\infty} \frac{(4 x-9) \cos \frac{\pi x}{5}+(-8 x-3) \sin \frac{\pi x}{5}}{x^{2}+4 x+40} d x \text {. } \]