I actually have a question on this one (love the book btw).
Usually, when computing E[g(x)], we use the density for X "f_X" and write E[G(x)] = int[ g(x)f_X dx]
Going back to this question I just wrote E[S_T^2]=int[ S_T^2 f_S dS] where f_S is the lognormal density for the asset, then moved to the log variables, etc..
However I took a peek at the solution to see if I was right (then I could just skip over the integration since it's mechanical) and I saw that the answer was quite far from my idea.
Could anyone give a quick hint at why my approach fails ?