How can I integrate the exponential $\int_{-\infty}^{\infty} e^{-au^2} du$?

Question

How can I integrate:

$$\int_{-\infty}^{\infty} e^{-au^2} du$$

Please note that I want to learn a step by step approach for doing so.

Answer 1

Let $$I = \int_{-\infty}^{\infty} e^{-au^2} du$$

Then

$$I = \int_{-\infty}^{\infty} e^{-av^2} dv$$

therefore $$ I^2 = \int_{-\infty}^{\infty} e^{-au^2} du\int_{-\infty}^{\infty} e^{-av^2} dv$$

$$ I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}e^{-au^2} du e^{-av^2} dv$$ $$ I^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}e^{-a(u^2+v^2)} du dv$$

Upon using polar coordinate, we get

$$I^2 = \int_{0}^{2\pi } \int_{0}^{\infty}e^{-a(r^2)} rdrd\theta =\frac {\pi}{a}$$

Thus $$I = \sqrt {\frac {\pi}{a}}$$