# Thermal Physics

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$$\int x^{2n} \textit{N}(x) dx$$

$$\int x^{2n} \textit{N}(x) dx$$
I suppose the inductive part comes from solving for the first few n and then coming up with a general n dependent solution. So yeah use integration by parts to find the first few moments. One can use the simpler method of observing that $$\int x^{2n}e^{-\alpha x^2} dx = (-1)^n \int \frac{\partial^{n}}{\partial \alpha^{n}} e^{-\alpha x^2} dx$$ and then using the standard results of a gaussian integral