\( \int e ^ { - \frac { 1 } { 2 } x ^ { 2 } } d x \)

Did you know that the integral \( \int e^{-\frac{1}{2}x^2} dx \) is closely related to the Gaussian function? This integral doesn't have a simple expression in terms of elementary functions, but it plays a crucial role in probability theory, particularly in the definition of the normal distribution. In fact, the area under the Gaussian curve is integral to understanding statistics and data analysis! In practical terms, this integral is often evaluated by using the concept of limits and properties of the error function, defined as \( \text{erf}(x) \). So while you could encounter it in various mathematical contexts, knowing how to relate it to the error function can save you a lot of headaches during calculations!