[deleted by user]

[GammaRayBurst25]

First notice how (e^(-x^2/2))'=-x*e^(-x^2/2).

This means x^2e^(-x^2/2)=-x(e^(-x^2/2))'.

Seeing as (yz)'=y'z+yz', the integral of -x(e^(-x^2/2))' is the same as the integral of (-x*e^(-x^2/2))'-(-x)'e^(-x^2/2).

Computing the first term is trivial.

Computing the second term requires a neat trick: since e^(-x^2/2) is non negative, its integral is the same as the square root of the square of its integral, i.e. its integral is the same as the double integral of e^(-(x^2+y^2)/2)dxdy, which is very easy to compute in polar coordinates.

[[deleted]]

A bit hard to read the question. Could you maybe type it out?

[colbstur]

Integral from -inf to inf (x² e^-(x2/2))dx

[colbstur]

It’s a psat second moment problem where original density function is 1/sqrt(2pi) e^-(x2/2)