# How to solve the integral of Ax^2 * exp(-x^2/2a^2)

What is the solution of equation?

intregal of Ax^2*exp(-x^2/2a^2)

lurflurf
Homework Helper
danai_pa said:

What is the solution of equation?

intregal of Ax^2*exp(-x^2/2a^2)
Do you want a definite or indefinite integral? The antiderivative of this function cannot be written in terms of elementry function. If you know the function erf(x) you can express the integral in terms of it (hint: integrate by parts). erf is usually defined in terms of the integral of exp(-x^2). You can also find the integral of the function over the whole real line by integrating by parts and using the fact that the whole real line integral of exp(-k x^2) is
sqrt(pi/k) a fact that can be shown a number of ways. How much calculus do you know?

If you are looking to integrate over the whole real line, and are familiar with statistics, another approach is to notice that, apart from a constant, this is similar to finding the expectation of x^2 where x is a normally distributed random variable with mean 0 and variance a^2. Using the formula

var[x] = E[x^2] - (E[x])^2