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Where X is normally distributed and f(x) is the PDF

How do I find the expectation of X

^{4}?

Bare with me because I'm useless in Latex

So far what I've done is written the integral as Integral between [-inf,inf] of X

^{4}*f(x) dx

and I started to subsitute u = exp{-x

^{2}/2t}

So now I have Integral between [-inf,inf] of -tX

^{3}*u dx

I really don't think this is correct.. I'm trying to follow the same way my lecturer did it for the expectation of X

^{2}, but at that statge he started to integrate by parts, yet mine doesn't look like that and it's essentially the same!

Can anyone help?

Thanks