standard normal expectations

Calculate $\displaystyle E[e^Z]$, where $\displaystyle Z$ is the standard normal random variable (with density function $\displaystyle f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$).

So am I to calculate $\displaystyle \int _{-\infty}^{\infty}e^{f(x)} \cdot f(x) \,\, dx$?

If I'm approaching this correctly, that leaves me with a mess stickier than a barrel of molasses which I don't know how to handle at all. I mean, I know that the integral of f(x) is F(x) where F(x) is the cumulative distribution function for the standard normal distribution, but I don't know what that is explicitly nor how to calculate it, let alone what is being asked here. In lecture, we have used tables of values for $\displaystyle \Phi (x)$, where $\displaystyle \Phi (x)$ (I assume) is this F(x), to approximate definite integrals of f(x) but I do not see how that might help here...so lost.

Suggestions, please?

A different interpretation

Your problem might be asking you to evaluate:

$\displaystyle

\int _{-\infty}^{\infty}e^{x} \cdot f(x) \,\, dx

$