9.2.26 Integrals depending on a parameter

Problem: Using the Dirichlet/Frullani/Fresnel/Euler/Poisson/Laplace integrals, calculate the following integral: \[ I=\int_{0}^{+\infty} \frac{\alpha \sin x-\sin \alpha x}{x^{2}} d x, \quad \alpha>0 . \]