Firepanda

I know the E[X] = Integral between [-inf,inf] of X*f(x) dx

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

How do I find the expectation of X⁴?

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 X⁴*f(x) dx

and I started to subsitute u = exp{-x²/2t}

So now I have Integral between [-inf,inf] of -tX³*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 X², 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