Basic question about an Integration example

I understand the basics of integration, but I'm extremely rusty, and in the process of refreshing myself, I hit an example I don't understand.

This is the problem:

$\displaystyle

\int \frac{2x}{1+x^2}dx

$

So, my understanding of the correct way to solve this is this:

$\displaystyle

2\int \frac{x}{1+x^2}dx

$

$\displaystyle

u = 1+x^2

$

$\displaystyle

du = 2xdx

$

$\displaystyle

=\int \frac{1}{u}du

$

$\displaystyle

=ln|u|+c

$

$\displaystyle

=ln|1+x^2|+c

$

Here's what I don't understand - what happened to $\displaystyle du$? $\displaystyle 2xdx$?

To me, it just dissappeared.