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# C.7.2 - Indefinite Integrals - PowerPoint PPT Presentation

C.7.2 - Indefinite Integrals. Calculus - Santowski. 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

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### C.7.2 - Indefinite Integrals

Calculus - Santowski

Calculus - Santowski

• 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

Calculus - Santowski

• 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

Calculus - Santowski

• 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

Calculus - Santowski

• Here is a list of common integrals:

Calculus - Santowski

• 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

Calculus - Santowski

• 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)

Calculus - Santowski

• (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 =

• (x2x)dx =

• (cos + 2sin3)d =

• (8x + sec2x)dx =

• (2 - x)2dx =

Calculus - Santowski

• Continue now with these questions on line

• Problems & Solutions with Antiderivatives / Indefinite Integrals from Visual Calculus

Calculus - Santowski

• 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

Calculus - Santowski

• 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

Calculus - Santowski

• 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

Calculus - Santowski

(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)

Calculus - Santowski