[Math] Integral of $e^{-x^{2}}$ and the error function

calculuserror functionintegration

How to integrate $e^{-x^{2}}$?

When I used geogebra I got the answer as $\frac{1}{2}\sqrt{\pi}\operatorname{erf}(x)$. What is $\operatorname{erf}(x)$ ?

How to arrive at this answer?

Best Answer

This integral cannot be expressed in terms of elementary functions such as polynomials, trigonometric, exponentials or logarithms. It is however a very important integral so mathematicians made up an answer and gave it a new name - $\operatorname{erf}(x)$. The definition of $\operatorname{erf}(x)$ is hence based on the integral. It full name is the Error Function. It is often encountered in statistics/probability and the solving of differential equations.

Related Question