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C.7.2 - Indefinite IntegralsPowerPoint Presentation

C.7.2 - Indefinite Integrals

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Lesson Objectives

- 1. Define an indefinite integral
- 2. Recognize the role of and determine the value of a constant of integration
- 3. Understand the notation of f(x)dx
- 4. Learn several basic properties of integrals
- 5. Integrate basic functions like power, exponential, simple trigonometric functions
- 6. Apply concepts of indefinite integrals to a real world problems

Calculus - Santowski

Fast Five

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(A) Review - Antiderivatives

- Recall that working with antiderivatives was simply our way of “working backwards”
- In determining antiderivatives, we were simply looking to find out what equation we started with in order to produce the derivative that was before us
- Ex. Find the antiderivative of a(t) = 3t - 6e2t

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(B) Indefinite Integrals - Definitions

- Definitions: an anti-derivative of f(x) is any function F(x) such that F`(x) = f(x) If F(x) is any anti-derivative of f(x) then the most general anti-derivative of f(x) is called an indefinite integral and denoted f(x)dx = F(x) + C where C is any constant
- In this definition the is called the integral symbol, f(x) is called the integrand, x is called the integration variable and the “C” is called the constant of integration So we can interpret the statement f(x)dx as “determine the integral of f(x) with respect to x”
- The process of finding an indefinite integral (or simply an integral) is called integration

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(D) Properties of Indefinite Integrals

- Constant Multiple rule:
- [c f(x)]dx = c f(x)dx and -f(x)dx = - f(x)dx
- Sum and Difference Rule:
- [f(x) + g(x)]dx = f(x)dx + g(x)dx
- which is similar to rules we have seen for derivatives

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(D) Properties of Indefinite Integrals

- And two other interesting “properties” need to be highlighted:
- Interpret what the following 2 statement mean:
- g`(x)dx = g(x) + C
- d/dx f(x)dx = f(x)

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(E) Examples

- (x4 + 3x – 9)dx = x4dx + 3 xdx - 9 dx
- (x4 + 3x – 9)dx = 1/5 x5 + 3/2 x2 – 9x + C
- e2xdx =
- sin(2x)dx =
- (x2x)dx =
- (cos + 2sin3)d =
- (8x + sec2x)dx =
- (2 - x)2dx =

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(F) Examples

- Continue now with these questions on line
- Problems & Solutions with Antiderivatives / Indefinite Integrals from Visual Calculus

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(G) Indefinite Integrals with Initial Conditions

- Given that f(x)dx = F(x) + C, we can determine a specific function if we knew what C was equal to so if we knew something about the function F(x), then we could solve for C
- Ex. Evaluate (x3 – 3x + 1)dx if F(0) = -2
- F(x) = x3dx - 3 xdx + dx = ¼x4 – 3/2x2 + x + C
- Since F(0) = -2 = ¼(0)4 – 3/2(0)2 + (0) + C
- So C = -2 and
- F(x) = ¼x4 – 3/2x2 + x - 2

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(H) Examples – Indefinite Integrals with Initial Conditions

- Problems & Solutions with Antiderivatives / Indefinite Integrals and Initial Conditions from Visual Calculus
- Motion Problem #1 with Antiderivatives / Indefinite Integrals from Visual Calculus
- Motion Problem #2 with Antiderivatives / Indefinite Integrals from Visual Calculus

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(I) Internet Links Conditions

- Calculus I (Math 2413) - Integrals from Paul Dawkins
- Tutorial: The Indefinite Integral from Stefan Waner's site "Everything for Calculus”
- The Indefinite Integral from PK Ving's Problems & Solutions for Calculus 1
- Karl's Calculus Tutor - Integration Using Your Rear View Mirror

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(J) Homework Conditions

- Textbook, p392-394
- (1) Algebra Practice: Q5-40 as needed + variety
- (2) Word problems: Q45-56 (economics)
- (3) Word problems: Q65-70 (motion)

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