About This Chapter
Once you've learned about derivatives and differential equations, it's time to study anti-derivatives and integrals. Watch these videos for an introduction to integrals and the integration techniques most often used by mathematicians in calculus. You'll find that many of the methods you use are the opposite of finding the derivative of variables that you've learned in other lessons.
First, learn how to use integration by parts. These can be used to solve complex product integrals. You may recognize that this is the reverse method of the product rule used when finding derivatives. In either case, you'll separate the trigonometric functions from the algebraic functions first. Then, learn how to calculate definite and indefinite integrals of trig functions, polynomials and exponentials.
Study the substitution technique, where you replace u for f(x). This technique is supposed to simplify equations so that you can integrate for u and then solve for f(x) once you've replaced it back in. It sounds confusing, but when watching these videos, you'll learn essential methods for solving difficult integrals and trigonometric substitution. Then, gain practice by watching multiple examples to ease your understanding.
Next, learn how to work with partial fractions inside functions. Discover how to recognize simple shapes by function, such as the trapezoid rule. And finally learn how to solve improper integrals. Then gain practice.
1. 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.
2. 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.
3. 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.
4. 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!
5. 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.
6. 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.
7. 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!
8. Partial Fractions: How to Factorize Fractions with Quadratic Denominators
Adding fractions with different denominators is something you probably learned how to do in algebra. In this lesson, learn how to do the opposite: take a complicated fraction and turn it into two simpler ones.
9. How to Integrate Functions With Partial Fractions
In this lesson, learn how to integrate complicated fractions by using the partial fractions technique. That is, you will turn a complicated fraction into something a bit easier to integrate by finding partial fractions!
10. Understanding Trigonometric Substitution
Sometimes a simple substitution can make life a lot easier. Imagine how nice it would be if you could replace your federal tax form with a 'Hello, my name is...' name badge! In this lesson, we review how you can use trigonometry to make substitutions to simplify integrals.
11. How to Use Trigonometric Substitution to Solve Integrals
In this lesson, we use each of the common integration techniques to solve different integrals. It's not always obvious which technique will be the easiest, so being familiar with an arsenal of methods might save you a lot of work!
12. How to Solve Improper Integrals
What does it mean when an integral has limits at infinity? These integrals are 'improper!' In this lesson, learn how to treat infinity as we study the so-called improper integrals.
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Other chapters within the Math 104: Calculus course
- Graphing and Functions
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Rate of Change
- Calculating Derivatives and Derivative Rules
- Graphing Derivatives and L'Hopital's Rule
- Applications of Derivatives
- Area Under the Curve and Integrals
- Integration Applications
- Differential Equations
- Studying for Math 104