E of a signal using Rayleigh
If I understand correctly, you want to verify the energy calculation in the frequency domain by computing the energy as

$$E_x=\int_{-\infty}^{\infty}|X(f)|^2df\tag{1}$$

with

$$X(f)=\mathcal{F}\big\{x(t)\big\}=\frac{A}{A+i2\pi f}\tag{2}$$

From $(2)$ we get

$$|X(f)|^2=\frac{A^2}{A^2+(2\pi f)^2}=\frac{1}{1+\left(\frac{2\pi f}{A}\right)^2}\tag{3}$$

With $(3)$ and with the substitution $x=2\pi f/A$, the integral $(1)$ becomes

$$E_x=\frac{A}{2\pi}\int_{-\infty}^{\infty}\frac{1}{1+x^2}dx=\frac{A}{2\pi}\arctan(x){\huge|}_{-\infty}^{\infty}=\frac{A}{2\pi}\cdot\pi=\frac{A}{2}\tag{4}$$