Integral of a Gaussian function

Question

I want to determien the following integral: $\int_\mathbb{R}|x-\dfrac{\sigma}{\sigma_{n}}x|g_{0,\sigma_{n}^{2}}(x)dx$, where $g_{0,\sigma_{n}^{2}}$ is a Gaussian function.

Thanks for your help!!

Answer 1

I assume that $$ g_{a,b}(x)=e^{-(x-a)^2/2b} $$then one may write $$ \begin{align} \int_\mathbb{R}\left|x-\dfrac{\sigma}{\sigma_{n}}x\right|g_{0,\sigma_{n}^{2}}(x)\:dx&=\left|1-\dfrac{\sigma}{\sigma_{n}}\right|\int_{-\infty}^\infty\,|x|\,e^{-x^2/2\sigma_{n}^{2}}\,dx \\\\&=2\left|1-\dfrac{\sigma}{\sigma_{n}}\right|\int_{0}^\infty\,x\,e^{-x^2/2\sigma_{n}^{2}}\,dx \\\\&=\left|1-\dfrac{\sigma}{\sigma_{n}}\right|\int_{0}^\infty\,e^{-u/2\sigma_{n}^{2}}\,du \quad (u=x^2,\, du=2xdx) \\\\&=\left|1-\dfrac{\sigma}{\sigma_{n}}\right|\cdot 2\sigma_{n}^{2} \\\\&=2\left|\sigma_{n}-\sigma\right| \sigma_{n}. \end{align} $$