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...
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