About This Chapter
Applying Integration Techniques - Chapter Summary
Better understand how to apply techniques for calculating integrals in various functions, use substitution, solve improper integrals and more with the video and text lessons comprising this chapter. Complex processes and applications are simplified through illustrations and examples so you will easily grasp each concept. You can expect to have a better handle on everything from anti-derivatives to trigonometric substitution.
In addition to simple instruction, you will enjoy 24/7 accessibility of all lessons, self-paced learning and the option to print lesson transcripts if needed. Test what you have learned with self-assessment quizzes to determine your readiness to apply these concepts on whatever exam you might be taking. Once you have completed this chapter, you should be able to:
- Find derivatives of logarithmic functions
- Calculate integrals of simple shapes, trigonometric functions and exponential functions
- Solve improper integrals and integrals using substitution
- Understand substitution techniques for difficult integrals
- Relate calculations used for indefinite integrals of polynomials
- Explain techniques for partial fractions
- Use integration by parts and trigonometric substitution to solve integrals
1. Calculating Derivatives of Logarithmic Functions
Logarithms appear in all types of applications like carbon dating, brightness of stars and musical acoustics. In this lesson we explore how to calculate the derivative of logarithmic functions.
2. 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.
3. 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.
4. 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.
5. 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!
6. 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.
7. 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.
8. 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!
9. 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.
10. 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!
11. 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.
12. 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!
13. 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 AP Calculus BC: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Function Continuity
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Approximation of Definite Integrals
- Understanding Sequences & Series
- Series of Constants
- Taylor Series
- Using a Scientific Calculator for Calculus
- AP Calculus BC Flashcards