Question

Solve the integral:20dx" id="TexFormula1" title=" \int \: {e}^{ - {x}^{2} } dx" alt=" \int \: {e}^{ - {x}^{2} } dx" align="absmiddle" class="latex-formula">

Answer 1

$e^{-x^2}$ has no antiderivative in terms of elementary functions (polynomials, exponentials, logarithms, trigonometric functions, etc), but there is a special function defined to fit that role called the error function, $\mathrm{erf}(x)$, where

$\mathrm{erf}(x)=\displaystyle\frac2{\sqrt\pi}\int_0^xe^{-t^2}\,\mathrm dt$

By the fundamental theorem of calculus, we can see that

$\dfrac{\mathrm d}{\mathrm dx}\mathrm{erf}(x)=\dfrac2{\sqrt\pi}e^{-x^2}$

which means we have

$\displaystyle\int e^{-x^2}\,\mathrm dx=\dfrac{\sqrt\pi}2\mathrm{erf}(x)+C$