= Solution
Differentiating $\int e^{-a x^2}dx=\sqrt{\pi/a}$ with respect to $a>0$ gives $\int x^2e^{-a x^2}dx=\sqrt\pi/(2a^{3/2})$. Therefore
$$
\boxed{E=\int_{-\infty}^{\infty}x^2e^{-2x^2}\,dx=\frac{\sqrt\pi}{4\sqrt2}.}
$$
From part (iii), $|\widetilde f(k)|^2=(\pi/4)k^2e^{-k^2/2}$. Its frequency-side energy is
$$
\frac1{2\pi}\int_{-\infty}^{\infty}|\widetilde f(k)|^2dk=\frac18\int_{-\infty}^{\infty}k^2e^{-k^2/2}dk=\frac{\sqrt{2\pi}}8=\frac{\sqrt\pi}{4\sqrt2},
$$
in agreement with <Parseval's identity>.
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