9.2.17 Integrals depending on a parameter

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