9.2.7 Integrals depending on a parameter

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