# Ch 11: Integration and Integration Techniques: Help and Review

### About This Chapter

## Who's it for?

Anyone who needs help understanding material from college calculus will benefit from taking this course. You will be able to grasp the subject matter faster, retain critical knowledge longer and earn better grades. You're in the right place if you:

- Have fallen behind in understanding integration or working with integration techniques.
- Need an efficient way to learn about integration and integration techniques.
- Learn best with engaging auditory and visual tools.
- Struggle with learning disabilities or learning differences, including autism and ADHD.
- Experience difficulty understanding your teachers.
- Missed class time and need to catch up.
- Can't access extra math learning resources at school.

## How it works:

- Start at the beginning, or identify the topics that you need help with.
- Watch and learn from fun videos, reviewing as needed.
- Refer to the video transcripts to reinforce your learning.
- Test your understanding of each lesson with short quizzes.
- Submit questions to one of our instructors for personalized support if you need extra help.
- Verify you're ready by completing the Integration and Integration Techniques chapter exam.

## Why it works:

**Study Efficiently:**Skip what you know, review what you don't.**Retain What You Learn:**Engaging animations and real-life examples make topics easy to grasp.**Be Ready on Test Day:**Use the Integration and Integration Techniques chapter exam to be prepared.**Get Extra Support:**Ask our subject-matter experts any relevant question. They're here to help!**Study With Flexibility:**Watch videos on any web-ready device.

## Students will review:

In this chapter, you'll learn the answer to questions including:

- How do I calculate integrals of simple shapes?
- What do I do to calculate integrals of polynomials?
- How do I calculate integrals of exponential or trigonometric functions?
- How do I use substitution to simplify complex integrals?
- What steps do I follow when factoring fractions with quadratic denominators?
- What is trigonometric substitution, and how is it used to calculate improper integrals?

### 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.

### 13. Initial Value in Calculus: Definition, Method & Example

In this lesson, you'lll learn about how the initial value of calculus is related to the integration process. You will see how crucial having an initial condition is and how useful it is in solving your problems.

### 14. Midpoint Rule: Formula & Example

Finding an estimate for the area under a curve is a task well-suited to the midpoint rule. In this lesson we use an example to show the general idea of this formula and how to use it.

### 15. How to Integrate xe^x: Steps & Tutorial

In this lesson, we use the product rule and integration by parts to find the integral of xe^x. The natural extension of integration by parts leads to a reduction formula which elegantly extends the integration results.

### 16. How to Integrate sec(5x): Steps & Tutorial

In this lesson you will learn how to integrate sec 5x using two different methods. After showing the results are equivalent, we will verify the solution and use it to find the area under the curve of sec 5x over a finite interval.

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### Other Chapters

Other chapters within the Calculus: Help and Review course

- Graphing and Functions: Help and Review
- Continuity in Calculus: Help and Review
- Geometry and Trigonometry in Calculus: Help and Review
- Using Scientific Calculators in Calculus: Help and Review
- Limits in Calculus: Help and Review
- Rate of Change in Calculus: Help and Review
- Calculating Derivatives and Derivative Rules: Help and Review
- Graphing Derivatives and L'Hopital's Rule: Help and Review
- Applications of Derivatives: Help and Review
- Area Under the Curve and Integrals: Help and Review
- Integration Applications: Help and Review
- Differential Equations: Help and Review