Q:

How do I handle $dx$ in $u$-substitution?

I am having trouble with an example of $u$-substitution:
$$ \int \frac{x}{x²+1}dx$$
In the next step they write:
Let $u = x^2+1$ which seems like a good choice. Then $du = 2x$ and that is also obvious, but then: $xdx = \frac{1}{2}$ and then use that to do something as I don't understand. 
How is the $xdx$ value calculated and what are they using it for? As you see, I am having trouble with understanding $u$-substitution in general....

A:

If you choose $u=x^2+1$, then taking the derivative with respect to $x$ gives:
$$\frac{\textrm{d}u}{\textrm{d}x}=2x.$$
Therefore $\textrm{d}u=2x\textrm{d}x$ or $x\textrm{d}x=\textrm{d}u/2$. Now your integral is written:
$$\frac{1}{2}\int\frac{1}{u}\textrm{d}u,$$
which is easier to solve than the original integral. It is equal to:
$$\frac{1}{2}\ln{|u|}\ =\, \frac{1}{2}\ln{(x^2+1)}.$$

