calculus - Showing that $int_{-infty}^{infty} exp(-x^2) ,mathrm{d}x = sqrt{pi}$

The primitive of $f(x) = \exp(-x^2)$ has no analytical expression, even so, it is possible to evaluate $\int f(x)$ along the whole real line with a few tricks. How can one show that

$$\int_{-\infty}^{\infty} \exp(-x^2) \,\mathrm{d}x = \sqrt{\pi} \space ?$$

Answer

Such an integral is called a Gaussian Integral

This link should help you out.