Finding Energy of Signal

Question

I'm trying to calculate the energy $E$ of the signal $e^{-t^2/{2\sigma^2}}$ using the energy formula:

$$E_t = \int_{-\infty}^{+\infty}|e^{-t^2/{2\sigma^2}}|^2dt$$

But i'm not how to start the integral .... any help would be great.

Answer 1

$$E_t = \int_{-\infty}^{+\infty}e^{-t^2/{\sigma^2}}dt$$

$$E^2 = \int_{-\infty}^{+\infty}e^{-y^2/{\sigma^2}}dy\int_{-\infty}^{+\infty}e^{-x^2/{\sigma^2}}dx$$

$$E^2 = \int_{-\infty}^{+\infty}e^{-y^2/{\sigma^2}}e^{-x^2/{\sigma^2}}dxdy$$

$$E^2 = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}e^{-\frac{x^2+y^2}{\sigma^2}}dxdy$$

$$E^2 = \int_{0}^{2\pi}\int_{0}^{+\infty}e^{-\frac{r^2}{\sigma^2}}rdrd\theta$$

$$2\pi \frac{\sigma^2}{2}$$

$$E = \sigma \sqrt{ \pi}$$