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## Homework Statement

Doing calculations for quantum mechanics, I have stumbled upon several integrals which I cannot solve by hand.

All of these have the form:

[tex]

\int x^n exp[-x^2] dx

[/tex]

For instance, when calculating the expectation value for x^2, the integral reduces to

[tex]

\int_{-\infty}^{\infty} x^2 exp[-x^2] dx

[/tex]

multiplied by some constant factors.

## Homework Equations

There is a standard integral given in the back of my book (Griffiths):

[tex]

\int_{0}^{\infty} exp[-\frac{x^2}{a^2}] dx = n! a^{n-1}

[/tex]

However, when I apply this to n=2, I arrive at a different answer than when I use Mathematica to compute it.

Also, when I try WolframAlpha, it returns a function that involves the error function without explaining how it got there.

## The Attempt at a Solution

There seems to be no way of simplifying or rearranging the integral. Integration by parts only works for taking the derivative of the exponential part, but that leaves higher and higher orders of n. Making substitution for x^2 still leaves a factor x in the denominator since u = x^2 implies du = 2 x dx. The result then would give a square root of u in the integral, which I cannot solve by integration by parts.

Is there any way to find an explicit formula for this integral?