Improper integral with parameters

Question

I need to find the values of $\alpha ,\beta \in \mathbb{R} $ so the integral :

$$Ι=\int_0^\infty \frac{(\alpha+\beta)+\beta}{2x^2+\beta}=\pi$$

I tried calculating the integral, but I do not know if $\beta$ is positive; any help would be appreciated.

Answer 1

$$\int_{0}^{\infty}\dfrac{\alpha+2\beta}{2x^2+\beta}\mathrm dx=\dfrac{1}{2}\cdot (\alpha+2\beta)\int_{0}^{\infty}\dfrac{1}{x^2+\beta/2}\mathrm dx=\dfrac{1}{2}\cdot (\alpha+2\beta)\cdot\left[\sqrt{\dfrac{2}{\beta}}\arctan\left(x\sqrt{\dfrac{2}{\beta}}\right)\right]_{0}^{\infty}$$

Can you proceed?