About This Chapter
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- Verify you're ready by completing the Integration and Integration Techniques chapter exam.
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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
Explore descriptions of how to calculate the definite integral of simple shapes through geometry. Learn how to calculate the integral of a simple shape, such as a rectangle or a trapezoid, find the area under the curve, and solve for the area.
2. Anti-Derivatives: Calculating Indefinite Integrals of Polynomials
The indefinite integral of ''f(x)dx'' can be written as the anti-derivative of ''f(x)'' plus some integration constant. Explore more about anti-derivatives and calculating indefinite integrals of polynomials with examples included in this lesson.
3. How to Calculate Integrals of Trigonometric Functions
Discover the two main integrals of trigonometric functions that are the most important to understand. Review how integrals are calculated and the integrals of sine and cosine and study examples of calculating these integrals.
4. How to Calculate Integrals of Exponential Functions
Taking the derivative of the function e^x results in e^x, while taking the indefinite integral of e^x dx equals e^x + C. Review examples of how to calculate integrals of exponential functions.
5. How to Solve Integrals Using Substitution
Consider how to solve integrals using substitution, which is a result of a tool called the chain rule. Review the chain rule and its uses in solving substitution, find examples of substitution, and discover how to check the answer to a problem.
6. Substitution Techniques for Difficult Integrals
Most integrals can be solved by looking them up in a table or by replacing part of the function with ''u'' substitution. Learn more about solving difficult integrals using substitution techniques.
7. Using Integration By Parts
Integration by parts takes the integral udv and separates it into uv and another integral that is easier to solve. Discover more about the mathematical concept and how it is similar to completing a jigsaw puzzle.
8. Partial Fractions: How to Factorize Fractions with Quadratic Denominators
Solving equations with partial fractions is made easier by factoring and determining what the undetermined coefficients A and B are. Learn how to factorize fractions that have quadratic denominators with problem examples.
9. How to Integrate Functions With Partial Fractions
Discover how to integrate complex functions with partial fractions. Learn about partial fractions and when to use them, when to use ~'u~' substitutions, and find examples of how to integrate functions using these techniques.
10. Understanding Trigonometric Substitution
Trigonometric substitution uses substitutions based on trigonometric identities. Gain an enhanced understanding of using trigonometric substitutions to simplify integrals by reviewing a series of examples.
11. How to Use Trigonometric Substitution to Solve Integrals
Solving problems of integration may involve memorizing certain integrals, such as sin(x) and 1/x, or may involve other tools like Riemann Sums. Explore more about solving integrals with trigonometric substitution.
12. How to Solve Improper Integrals
Improper integrals have either a positive or negative infinity. Explore how to solve them by splitting up the integrals and replacing infinity with a variable.
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 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