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

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Presentation transcript:

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 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.

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

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 “S” symbol comes from the word “Sum” and is the integral sign. constantintegrand The differential

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.

Definition of Indefinite Integration:

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!

Integral of a Sum of Two Functions: integration distributes over addition.

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

Integrating Composite Functions Using U- Substitution:

Property:

Integration of Example : Evaluate the indefinite integral

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

Properties: Derivative and Integral of an Exponential Function Ω