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I presume you have a Gaussian RV with mean µ, and variance σ².
The integral with |x| is awkward to think of - break it into two pieces.
With a little thought, the integral from 0 to ∞ is always positive - no need for |x|.
Similarly, the integral from -∞ to 0 is always negative, but that's exactly where
|x| = -x. Your answer is the difference between the two integrals.
(the integral from 0 to ∞, minus the integral from -∞ to 0)
Using integral tables and whatnot, the answer is a combination of exponentials and the error functions erf(). Evaluating the limits approaching infinity has a pair of the exponentials going to 0, a pair of erf() going to ±1 (canceling).
In the end you have the sum of two terms - one an exponential, and one an erf().
(σ√(2/Π))×exp(-(µ²/(2σ²))) + µ×erf(µ/(σ√2))
The first √ is of 2/Π - sorry for this sloppy mathematical notation.
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