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Is there any closed form expression for the following integral?

$ \int\limits_t^\infty \left(1- \operatorname{erf}(\log x) \right )dx $

or equivalently:

$ \int\limits_t^\infty \operatorname{erfc}(\log x ) dx $

I just wish to know if there is any way I can do better than sampling the values at particular points and calculating the area under the curve numerically. Any help is appreciated. Thanks.

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    hey, thanks for your help. By the way, I tried to compute it symbolically in MATLAB, but it did not work out. I just hope that the given formula works. :)2012-04-10

1 Answers 1

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Not terribly hard; one only needs simple substitutions and an application of integration by parts here. To wit,

$\begin{align*} \int_t^\infty \mathrm{erfc}(\log u)\,\mathrm du&=\int_{\log\,t}^\infty \exp\,v\;\mathrm{erfc}\,v\,\mathrm dv\\ &=\left.\exp\,v\;\mathrm{erfc}\,v\right|_{\log\,t}^\infty+\frac2{\sqrt\pi}\int_{\log\,t}^\infty \exp\,(v-v^2)\,\mathrm dv\\ &=-t\;\mathrm{erfc}(\log\,t)+\frac{2\sqrt[4]{e}}{\sqrt\pi}\int_{\log\,t-\frac12}^\infty \exp\,(-w^2)\,\mathrm dw\\ &=\sqrt[4]{e}\,\mathrm{erfc}\left(\log\,t-\frac12\right)-t\;\mathrm{erfc}(\log\,t)\\ \end{align*}$

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    thanks for the step by step derivation.2012-04-10