Solving $\int_0^{+\infty}\frac{e^{-\alpha x^2} - \cos{\beta x}}{x^2}dx$ I need to find solution of $$\int_0^{+\infty}\frac{e^{-\alpha x^2} - \cos{\beta x}}{x^2}dx$$
I know that Leibniz rule can help but I don't know how to use it.
Could you help me please?
Thank you.

Comparison with $1/x$ shows that the integral does not converge at $x=0$.
