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C.7.2 - Indefinite Integrals

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

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  1. C.7.2 - Indefinite Integrals Calculus - Santowski Calculus - Santowski

  2. 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

  3. Fast Five Calculus - Santowski

  4. (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 Calculus - Santowski

  5. (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 Calculus - Santowski

  6. (C) Review - Common Integrals • Here is a list of common integrals: Calculus - Santowski

  7. (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 Calculus - Santowski

  8. (D) Properties of Indefinite Integrals • And two other interesting “properties” need to be highlighted: • Interpret what the following 2 statement mean: Use your TI-89 to help you with these 2 questions • Let f(x) = x3 - 2x • What is the answer for  f `(x)dx ….? • What is the answer for d/dx f(x)dx ….. ? Calculus - Santowski

  9. (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 = • (x2x)dx = • (cos + 2sin3)d = • (8x + sec2x)dx = • (2 - x)2dx = Calculus - Santowski

  10. (F) Examples • Continue now with these questions on line • Problems & Solutions with Antiderivatives / Indefinite Integrals from Visual Calculus Calculus - Santowski

  11. (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 Calculus - Santowski

  12. (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 Calculus - Santowski

  13. (I) Examples with Motion • An object moves along a co-ordinate line with a velocity v(t) = 2 - 3t + t2 meters/sec. Its initial position is 2 m to the right of the origin. • (a) Determine the position of the object 4 seconds later • (b) Determine the total distance traveled in the first 4 seconds Calculus - Santowski

  14. (J) Examples – “B” Levels • Sometimes, the product rule for differentiation can be used to find an antiderivative that is not obvious by inspection • So, by differentiating y = xlnx, find an antiderivative for y = lnx • Repeat for y = xex and y = xsinx Calculus - Santowski

  15. (K) Internet Links • 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

  16. (L) Homework • Textbook, p392-394 • (1) Algebra Practice: Q5-40 (AN+V) • (2) Word problems: Q45-56 (economics) • (3) Word problems: Q65-70 (motion) Calculus - Santowski

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