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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 dy 3 x 2dx We know 3 3x y c 3 y x c 3 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 dy 3 x 2dx Is the same as evaluating 3x dx 2 integrand • 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: g ( x ) f ( x )dx if and only if g '( x ) f ( x ) g ( x) x g '( x) 4 x 3 4 f ( x) 4 x 3 4 4x c 4x dx 4 3 x c 4 g ( x) f ( x)dx Two properties of Indefinite Integrals: • Integral of a Constant times a Function: kf ( x)dx k f ( x)dx constants can be pulled through the integral sign. Beware the variable can NOT! 1 4 3 t dt 4 3 1 t 3 dt 3 4 4t 3 c t 3 c 3 4 3 4 Two properties of Indefinite Integrals: • Integral of a Sum of Two Functions: f ( x) g ( x)dx f ( x)dx g ( x)dx integration distributes over addition. 3sin x sin 3xdx 3sin xdx sin 3xdx 3 sin xdx sin 3xdx cos3x 3cos x c 1 3 3cos x cos3 x c 3 Integrating Composite Functions Using USubstitution: • Power Rule for Integration: n≠-1 u u du c n 1 ( x 2) 6 u u du c 6 5 n 1 n 5 dx Let u = x+2 1 6 ( x 2) c 6 Integrating Composite Functions Using USubstitution: 4 x 1dx 3 2 Let u 4 x 1 (4 x 1) 2 c 3 3 2 1 3 2 u 2 2 ( u ) c c u du 3 3 2 Integrating Composite Functions Using USubstitution: Let u 4 x 1 Since u 4 x 1 u ' 4dx 1 So u dx 4 4 x 1dx 3 2 (4 x 1) 2 c 3 1 4 x 1dx u du 4 1 2 3 1 u2 c 4 32 Integrating Composite Functions Using USubstitution: 4 x 1dx 3 2 1 12 1 2 32 1 u u du c (u ) c 4 4 3 4 32 1 32 (u ) c 1 (4 x 1) 3 2 c 6 6 3 2 (4 x 1) 2 c 3 1 3 (4 x 1) c 6 Property: If f ( x) e and f '( x) e then x x e dx e c x x Integration of e x Example : Evaluate the indefinite integral 6 x 4 e 4 e 6 x dx dx 2 6 x e c 3 let u 6 x u ' 6dx 1 4 eu du 6 4 u e c 6 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 b e x 2 e x x ln b x ln 2 x x ln 2 2 dx e dx let u x ln 2 du ln 2dx 1 1 u u e du e c ln 2 ln 2 x ln 2 x e 2 c c ln 2 ln 2 Properties: Derivative and Integral of an Exponential Function d x x b b ln b dx 1 b dx b ln b c x x Ω