Given $X \sim N(1,2)$, what is $E[X^3]$?

I've been told I should use the moment generating function, but I'm not sure how to apply it here in this instance.

Using the given mean and variance, I subbed the values into the normal density function $f(x)$, then tried to compute $E[X^3]$ by integrating $\int\_{-\infty}^{\infty} f(x) \, x^3 \mathrm{d}x$ . At this point, I have no idea how to compute the integral, which leaves me pondering if I've veered way off track. I'd appreciate any help!