How do you integrate $E$ to find motion? Can we see an example?

If you know $U(x)$, either numerically or analytically, you can plop it in to $t-t_0 = \int_{x_0}^x \frac{\pm dx'}{\sqrt{\frac{2}{m}[E- U(x')]}}$ and do the integral (analytically if you can, or numerically. Note that the integration variable should be a dummy variable). This will give you a function of $x$ on the right and $t$ on the left, so you can find $x(t)$ which describes the motion. Example 4.6 in your text solves the free fall problem in this way, for $U=-mgx$. We'll be seeing more examples.