$x=4$

The formula $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-x^2/2}dx = 1$ is one of the most important improper integral calculations in mathematics. It is usually first met in probability theory, where it leads to the normal distribution.There are many proofs of this basic result. We will see how many I can get through before time runs out.