Evaluation of $dx$ in trigonometric substitution

Question

In evaluating integrals through trig substitution, I often see $dx$ get evaluated to something when substituting $x$ for something in terms of $\theta$. For example, in Sal Khan's explanation of $$\int \frac{1}{\sqrt{4-x^2}}dx$$ he evaluates $x=2sin\theta$ using the relationships between sides in a right triangle, which makes perfect sense to me, but then he sets $dx=2cos\theta d\theta$ without any explanation as to why that is mathematically valid or how you can do that. I was always of the understanding that $dx$ simply represented the variable of integration and was relatively immutable. How does this work, and why is it permissible?

Answer 1

\begin{align} x&=2\sin\theta\\ \frac{d}{d\theta}(x)&=\frac{d}{d\theta}(2\sin\theta)\\ \frac{dx}{d\theta}&=2\cos\theta\\ dx&=2\cos\theta\ d\theta \end{align}

Treating the $d\theta$ like a denominator is almost always permissible when working with single variable calculus.