Integration by substitution I'm attempting to learn Integration by substitution. However, I'm having trouble understanding how the $ \frac{dy}{dx} $ notation is being manipulated. I always understood that it was just notation and it shouldn't be treated as a fraction.
Consider the following integral: $$\int18x^2\sqrt[3]{6x^3+5}$$
In Paul's math notes, it then follows to let $$ u = 6x^3 + 5 $$ $$ \frac{du}{dx} = 18x $$
then this is the step, which I doubt the legitimacy and don't understand why it was done: $$du = 18x^2 dx$$
It then concludes the integral can be written as, $$ \int (6x^3+5)^\frac{1}{4}(18x^2dx)  $$ $$ = \int u^\frac{1}{4} du $$
I didn't understand any of that... 
 A: First, $\frac{du}{dx} = 18x^2$ (you lost the exponent). 
Second, your first integral is missing a "dx" at the end. I like to think of the integral sign and the "dx" as like open- and close-parentheses: they have to match up or what you've written doesn't make sense. 
The legitimacy of that step is that it's a mechanical process by which you end up at the same result as you do by applying the inverse of the chain rule. This isn't usually "proved" so much as shown in multiple cases to work, and then we tend to believe it'll work in all other similar-enough cases. To "prove" it is tough, since it's like proving something about parentheses -- this is really all about notation. 
So that's not really an answer to your question, but the point is that this is a mechanical process that you can learn to perform, and it gets you to the right answer. When you learned to expand
$$
(3x + 2) (4x - 1)
$$
you probably learned something like "FOIL" (first, outer, inner, last), meaning that you sum up the product of the first terms of each factor, 
$3x \cdot 4x$, the outer terms ($3x \cdot -1$), the inner terms $(2 \cdot 4x$) and the last terms $(2 \cdot -1$). Deep inside that rule is the application of the distributive and associative laws, but do you think about those every time you multiply out terms liek this? Of course not. The same thing goes here: this is a mechanical implementation of the consequences of the theorem that
$$
\int_a^b f(g(x)) g'(x) dx = \int_{g(a)}^{g(b)} f(u) du
$$
which is a consequence of the fundamental theorem of calculus and the chain rule. 
If you look at that theorem, you can see that each place there's a $u$ in the integrand on the right, there's a $g(x)$ on the left, and where, on the left, there's a $g'(x)dx$, there's a $du$ on the right. The rule you've been given simply implements this symbolic substitution. 
