generalisation of Gaussian integral
The integral
is a generalisation of the Gaussian integral . For evaluating it we first form its derivative which may be done by differentiating under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign):
Using integration by parts this yields
Thus satisfies the linear differential equation
where one can separate the variables (http://planetmath.org/SeparationOfVariables) and integrate:
So, , i.e. , and since there is the initial condition , we obtain the result