This is a long chapter, but it's gonna be worth it because this is a make-or-break skill that you'll be using throughout the rest of calculus. It will just be something you always have to do, sort of like the Chain Rule when you're taking derivatives. It's harder than the Chain Rule, though, so don't take it lightly!
What the heck is U-Substitution
U-Sub is really the Chain Rule in reverse, but it's difficult and more abstract. So in this video I work a couple examples, but mostly it's about showing you how U-sub is reverse chain rule, and how that can help you to understand what's going on.
How your teacher will make you show your work
There are a few different ways to work U-Substitution problems. They all get the same result, yet many calculus teachers real sticklers for you writing the steps exactly how they want you to. So my job, in this video, is to show you the most common ways of doing this, so that you'll be able to do it just like your teacher wants you to!
U-Substitution & The Power Rule
The most common type of U-sub problem is the power rule, since power rules come in so many different flavors: polynomials, exponents, fractions, roots... So this video takes you through examples of all the types, and some hard ones too, to get you used to this crazy U-sub stuff. Practice practice practice!
U-Substitution with Roots & Radicals
These are really just a sub-genre of the Power Rule, but they do have a lot of different algebra due to all the fractional exponents, so they get their own video! Including algebraic tricks to use when your u and du "don't match".
U-Substitution of Rational Functions
Rational functions are the big fractions with x's upstairs and down, but they come in many flavors. In this video I work rational examples with the Power Rule, arcsin and arctan, and tougher algebraic problems requiring tricky tricks before you can integrate. This video even covers a couple integrations that require synthetic division and polynomial long division!
U-Substitution of Trig Functions
Usually in U-sub problems, there's not a lot of drama in picking your u and du. It's fairly obvious. But trig functions are extra-tricky because teachers design problems that can be worked more than one way, or that take a lot of cleverness in picking your u and du to even get them to work. I also work a few examples that require the trig identities you'll need for calculus.
U-Substitution of Logs & Exponentials
We've already hit some ln|u| integrals in prior videos, but in this one we'll cover tricky problems where ln is inside a Power Rule problem, or exponentials are within rational functions. These are hard, but they'll take your game to the next level!
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