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.

You can separate the fraction, but it seems as Julian Aguirre pointed out, the first part does not converge at x=0:
$$
\int_0^{\infty}\dfrac{\exp(-α\,x)}{x^2}-\dfrac{\cosβ\,x}{x^2}\,dx
$$
This can be easily understood in the case of $α=0$.
But the indefinite integral can be found, it involves the incomplete gamma function.
Edit:
As you edited your question it is now solvable and indeed $\frac{\pi\,\beta}{2}-\sqrt{\alpha\pi}$ for $\beta>0$ and $\alpha>0$.
