calculus - 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...