# How to calculate the expected value of a standard normal distribution?

I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is where $f(x)$ is the probability density function of $X$.

Suppose the probability density function of $X$ is which is the density of the standard normal distribution.

So, I would first plug in the PDF and get

which is a rather messy looking equation. The constant $\displaystyle\frac{1}{\sqrt{2\pi}}$ can be moved outside the integral, giving

I get stuck here. How do I calculate integral? Am I doing this correctly this far? Is the simplest way to get the expected value?

Or you can directly use the fact that $xe^{-x^2/2}$ is an odd function and the limits of the integral are symmetry.