Primitive of Error Function

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$\ds \int \map \erf x \rd x = x \map \erf x + \frac 1 {\sqrt \pi} e^{-x^2} + C$

where $\erf$ denotes the error function.


By Derivative of Error Function, we have:

$\dfrac \d {\d x} \paren {\map \erf x} = \dfrac 2 {\sqrt \pi} e^{-x^2}$


\(\ds \int \map \erf x \rd x\) \(=\) \(\ds \int 1 \times \map \erf x \rd x\)
\(\ds \) \(=\) \(\ds x \map \erf x - \frac 2 {\sqrt \pi} \int x e^{-x^2} \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds x \map \erf x + \frac 1 {\sqrt \pi} \int \paren {-2 x e^{-x^2} } \rd x\)
\(\ds \) \(=\) \(\ds x \map \erf x + \frac 1 {\sqrt \pi} \int e^u \rd u\) substituting $u = -x^2$
\(\ds \) \(=\) \(\ds x \map \erf x + \frac 1 {\sqrt \pi} e^u + C\) Primitive of Exponential Function
\(\ds \) \(=\) \(\ds x \map \erf x + \frac 1 {\sqrt \pi} e^{-x^2} + C\) substituting back for $u$