1 / 17

Formal Definition of Antiderivative and Indefinite Integral

Formal Definition of Antiderivative and Indefinite Integral. Lesson 5-3. Back in chapter 1 we were given an outline of the topics of Calculus: Limits Derivatives One Type of Integral (definite integral) the Other Integral. Definite Integral .

pepin
Download Presentation

Formal Definition of Antiderivative and Indefinite Integral

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Formal Definition of Antiderivative and Indefinite Integral Lesson 5-3

  2. Back in chapter 1 we were given an outline of the topics of Calculus: • Limits • Derivatives • One Type of Integral (definite integral) • the Other Integral

  3. Definite Integral • The process of evaluating a product in which one factor varies. • Graphically, the area under the curve. Back in lesson 1-4 we learned about using trapezoids to approximate this area and applied the trapezoid rule.

  4. The Antiderivative is just another name for the Indefinite Integral. It is considered “Indefinite” because of the “+ C”, which we don’t know until we are given a starting point relative to our function. So, when asked to find an equation for the antiderivative We know or

  5. What we didn’t know at the time, when finding antiderivative, was that we were writing the answers to the indefinite integrals. The antiderivative of Is the same as evaluating The differential integrand The “S” symbol comes from the word “Sum” and is the integral sign. constant

  6. The purpose of discussing the differential in the previous lesson was because there is a relationship between it and the integral. • It tells us what variable we are performing the integration with respect to. • It comes in handy when dealing with those composite functions.

  7. Definition of Indefinite Integration: Given: So,

  8. Two properties of Indefinite Integrals: • Integral of a Constant times a Function: constants can be pulled through the integral sign. Beware the variable can NOT! Example: Evaluate

  9. Two properties of Indefinite Integrals: • Integral of a Sum of Two Functions: integration distributes over addition. Example: Evaluate

  10. Integrating Composite Functions Using U-Substitution: • Power Rule for Integration: n≠-1 Example: Evaluate Let u = x+2

  11. Integrating Composite Functions Using U-Substitution: Example: Evaluate a composite function is more complex

  12. Integrating Composite Functions Using U-Substitution: Example: Evaluate

  13. Integrating Composite Functions Using U-Substitution: Example: Evaluate Or Not

  14. Property:

  15. Integration of Example : Evaluate the indefinite integral

  16. Back in lesson 3-9 we learned to differentiate the exponential function 2x. With the following definition we can also integrate this exponential with a base other than e. Definition: Exponential with Base b So...

  17. Properties: Derivative and Integral of an Exponential Function Ω

More Related