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f(x) = 3x^2 for x in [0, 1]

f(x) = 0 elsewhere

I want to find E[X^2] which is easy if I use the integral definition (I got 3/5). Yet, when I try and do this using Moment Generating Function (mgf) I cannot seem to get the same answer (in fact I get an unbounded answer). I'm wondering why this is the case. Any help?