"Calculate the Fourier transform of $f(x)=\frac{1}{x^2+4}$ and verify that
$\int_{R} |f(x)|^2 dx = \int_R |\hat{f}(p)|^2dp$.
If I did it right, $ \hat{f}(p) = \frac{1}{\sqrt{2\pi}} \frac{\pi}{2}
\begin{cases}
e^{-2p} & \quad \text{if } p > 0\\
e^{2p} & \quad \text{if } p < 0
\end{cases}
$
Please let me know if I made mistakes.
For the second part, I don't know how to proceed. I would start stating that both $f(x)$ and $\hat{f}(p)$ $\in L^2(R) $. Now I know from lessons, but don't know how to prove it, that the energy of those functions is the same.
We have
$$\begin{align} \int_{-\infty}^\infty \frac{1}{(x^2+4)^2}\,dx&=2\int_0^{\pi/2}\frac{2\sec^2(x)}{16\sec^4(x)}\,dx\\\\ &=\frac14 \int_0^{\pi/2}\cos^2(x)\,dx\\\\ &=\frac\pi{16} \end{align}$$
and
$$\begin{align} \int_{-\infty}^\infty \frac\pi8 e^{-4|p|}\,dp&=\int_0^\infty \frac\pi4 e^{-4p}\,dp\\\\ &=\frac\pi 4\left.\left(\frac{e^{-4p}}{-4}\right)\right|_{p=0}^{p\to \infty}\\\\ &=\frac{\pi}{16} \end{align}$$
as was to be shown!