Expectation of the normal exponential martingale

With $X$ standard normal and $\theta$ a real constant, define $Z=\exp\!\left(\theta X-\tfrac12\theta^2\right)$. Compute $\mathbb{E}[Z]$.

$1$. The normal MGF gives $\mathbb{E}e^{\theta X}=e^{\theta^2/2}$, which exactly cancels the factor $e^{-\theta^2/2}$. This is the Radon-Nikodym density used in Girsanov changes of measure.