Fastest way to solve Gaussian Integral

Question

The Gaussian integral is defined as $$\int_0^\infty e^{-x^2}dx.$$

I know there are many ways to solve it, but what would be the fastest you know?

Answer 1

First of all the gaussian integral is defined as $$\int_{-\infty}^{\infty}e^{-x^2}\mathrm{d}x,$$ but since the function we want to integrate is even, we can easily find the connection between those two. The fastest way I know, is to use Eulers Reflection formula. I would start with the substitiution $x^2 = u$. That means that $\mathrm{d}x = \frac{\mathrm{d}u}{2\sqrt{u}}$. We get: $$\int_{-\infty}^{\infty}e^{-x^2}\mathrm{d}x = 2\int_{0}^{\infty}\cfrac{u^{-\frac{1}{2}}}{2}e^{-u}\mathrm{d}u.$$

If you are familiar with the gamma function you'll see that this integral is equivalent to $\Gamma(\frac{1}{2})$. Now use the Euler Reflection formula $$\Gamma(z)\Gamma(1-z) = \cfrac{\pi}{\sin{\pi z}}\quad{\Longrightarrow}\quad\Gamma\left(\frac{1}{2}\right)\Gamma\left(1-\frac{1}{2}\right) = \cfrac{\pi}{\sin{\frac{\pi}{2}}} $$ Now simplify and take the square root on both sides: $$\int_{-\infty}^{\infty}e^{-x^2}\mathrm{d}x = \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$$

Answer 2

what would be the fastest you know?

given that

$$\int_{-\infty}^{+\infty}e^{-x^2}dx=\sqrt{\pi}$$

...your integral is exactly the half: $\frac{\sqrt{\pi}}{2}$