Understanding the differential $dx$ when doing $u$-substitution I just finished taking my first year of calculus in college and I passed with an A.  I don't think, however, that I ever really understood the entire $\frac{dy}{dx}$ notation (so I just focused on using $u'$), and now that I'm going to be starting calculus 2 and learning newer integration techniques, I feel that understanding the differential in an integral is important.
Take this problem for an example:
$$ \int 2x (x^2+4)^{100}dx $$
So solving this...
$$u = x^2 + 4 \implies du = 2x dx \longleftarrow \text{why $dx$ here?}$$
And so now I'd have: $\displaystyle \int (u)^{100}du$ which is $\displaystyle \frac{u^{101}}{101} + C$ and then I'd just substitute back in my $u$.
I'm so confused by all of this.  I know how to do it from practice, but I don't understand what is really happening here.  What happens to the $du$ between rewriting the integral and taking the anti-derivative?  Why is writing the $dx$ so important?  How should I be viewing this in when seeing an integral?

$dx$ is what is known as a differential. It is an infinitesimally small interval of $x$:
$$dx=\lim_{x\to{x_0}}x-x_0$$
Using this definition, it is clear from the definition of the derivative why $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$:
$$y'=f'(x)=\frac{dy}{dx}=\lim_{x\to{x_0}}\frac{f(x-x_0)-f(x_0)}{x-x_0}$$
When doing u-substitution, you are defining $u$ to be an expression dependent on $x$. $\frac{du}{dx}$ is a fraction like any other, so to get $du$ you must multiply both sides of the equation by $dx$:
$$u=x^2+4$$
$$\frac{du}{dx}=2x$$
$$dx\frac{du}{dx}=2x\space{dx}$$
$$du=2x\space{dx}$$
