Originally Posted by

**Prove It** It's a sloppy shorthand, I'll give you that.

Really, what happens is that when you replace $\displaystyle \displaystyle \begin{align*} \sec^2{x} \end{align*}$ with $\displaystyle \displaystyle \begin{align*} \frac{du}{dx} \end{align*}$, in your integral you get

$\displaystyle \displaystyle \begin{align*} \int{\frac{\sec^2{x}}{(1 + \tan{x})^3}\,dx} &= \int{\frac{1}{u^3}\,\frac{du}{dx}\,dx} \\ &= \int{\frac{1}{u^3}\,du} \end{align*}$

which simplifies from the Chain Rule (not from "cancelling the $\displaystyle \displaystyle \begin{align*} dx \end{align*}$"). However, most people do the sloppy simplification to save time.