About This Chapter
Integrals & Antiderivatives - Chapter Summary
This chapter covers mathematic principles, calculations, and methods relevant to the presence and application of integrals and antiderivatives. You'll learn about topics such as finding the arc length of a function, solving integrals using substitution, using integration by parts, as well as all of the following:
- Indefinite integrals in relation to antiderivatives
- Using sigma notation
- Indefinite integrals of polynomials
- Calculating integrals of simple shapes, trigonometric functions, and exponential functions
- Substitution techniques for difficult integrals
Within each lesson you'll find the option of receiving the content through a video or transcript, enabling you to choose the learning style that works best for you. The timeline feature underneath the videos allows you to hop around the lesson's main points, while readers of the transcripts will find bolded words throughout, making it easy to spot key terms.
1. Indefinite Integrals as Anti Derivatives
What does an anti-derivative have to do with a derivative? Is a definite integral a self-confident version of an indefinite integral? Learn how to define these in this lesson.
2. How to Find the Arc Length of a Function
You don't always walk in a straight line. Sometimes, you want to know the distance between two points when the path is curved. In this lesson, you'll learn about finding the length of a curve.
3. Using Sigma Notation for the Sum of a Series
In math, we have a notation for a lot of things. When you need to sum up a series, we also have a notation for that. You will learn about this notation, called Sigma notation, in this video lesson.
4. Calculating Integrals of Simple Shapes
So you can write something called an integral with a weird squiggly line. Now what? In this lesson, calculate first integrals using your knowledge of nothing but geometry.
5. Anti-Derivatives: Calculating Indefinite Integrals of Polynomials
If you throw a ball in the air, the path that it takes is a polynomial. In this lesson, learn how to integrate these fantastic functions by putting together your knowledge of the fundamental theorem of calculus and your ability to differentiate polynomial functions.
6. How to Calculate Integrals of Trigonometric Functions
Ever feel like you are going around in circles? Like, periodically you have your ups and downs? Well, sines and cosines go up and down regularly too. In this lesson, learn how to integrate these circular functions.
7. How to Calculate Integrals of Exponential Functions
Exponential functions are so predictable. It doesn't matter how many times you differentiate e^x, it always stays the same. In this lesson, learn what this means for finding the integrals of such boring functions!
8. How to Solve Integrals Using Substitution
Some integrals are as easy as riding a bike. But more often, integrals can look like deformed bikes from Mars in the year 3000. In this lesson, you will learn how to transform these scary-looking integrals into simpler ones that really are as easy as riding a bike.
9. Substitution Techniques for Difficult Integrals
Up, down. East, West. Opposites are everywhere. In this lesson, learn how you can think of substitution for integration as the opposite of the chain rule of differentiation.
10. Using Integration By Parts
Your mother may have warned you not to bite off more than you can chew. The same thing is true with integration. In this lesson, learn how integration by parts can help you split a big interval into bite-sized pieces!
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