Integral of a Gaussian function
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The integral of an arbitrary Gaussian function is
An alternative form is
where f must be strictly positive for the integral to converge.
for some real constants a, b, c > 0 can be calculated by putting it into the form of a Gaussian integral. First, the constant a can simply be factored out of the integral. Next, the variable of integration is changed from x to y = x + b.
and then to
Then, using the Gaussian integral identity