About This Chapter
Indefinite Integrals in Calculus - Chapter Summary
This chapter holds indefinite integrals as its focus. You will find instruction on calculating integrals of trigonometric and exponential functions as well as integrating with partial fractions. If you are unfamiliar with integrals with infinite limits, the lesson on improper integrals could be helpful. Our lessons also discuss the various aspects and uses for substitution when solving integrals.
Here you'll find step-by-step instructions on calculating integrals in many forms. The primary focus of these tricks of the trade is simplification: picking apart the hard parts of your equation and substituting your way to a simple solution. Among the things you will learn to do in this chapter are:
- Define indefinite articles
- Calculate integrals in geometry and trigonometry
- Solve integrals with substitution
- Integrate partial fraction functions
- Understand trigonometric substitution
- Work with improper integrals
Our brief video lessons are taught by math experts who use humorous and entertaining graphics to help keep your attention and real-world examples to apply what you learn. Each video is accompanied by a lesson transcript so you can review key terms and see the examples in full form. The quizzes after each lesson can help you determine how much of the information you absorbed. Use the end-of-chapter test to measure how well you have synthesized the information and get an idea of where your strengths and weaknesses are for further review.
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. 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. 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 GRE Math: Study Guide & Test Prep course
- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
- GRE Math Flashcards