I'm trying to solve an improper integral :
$$\int_{-\infty}^{+\infty}\omega|\hat{G}(\omega)|^2 \, d\omega$$
where the function is it:
$$ G(x) = \begin{cases} 2-\frac{2x^2}{\pi^2} & -\pi<x \leq0 \\ cosx+1 & 0<x \leq \pi \end{cases} $$
I thought about fixing it using the Plancherel's theorem. But there is a problem that is , there is an $\omega$ before of module. Then i think that to solve this integral, i must to do a derivative of function G(x) than integrate the module of the derivative . Is the reasoning correct? Thank you so much