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\begin{equation}  I_1(\mu, \sigma) = \frac{\mu}{2} + \int_{0}^{\infty} df' f' \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{f'^2}{2\sigma^2} \right)  \end{equation}  Substituting $f'^2/(2\sigma^2)=t$, $f'df'/\sigma = \sigma dt$, we obtain  \begin{equation}  I_1(\mu, \sigma) = \frac{\mu}{2} + \int_{0}^{\infty} \frac{\sigma dt}{\sqrt{2\pi}}\exp(-t) = \frac{\mu}{2} + \frac{\sigma}{\sqrt{2\pi}}   \end{equation}