1. Calculus - Santowski 12/8/2015 Calculus - Santowski 1 C.7.2 - Indefinite Integrals

2. Lesson Objectives 12/8/2015 Calculus - Santowski 2 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

3. Fast Five 12/8/2015 Calculus - Santowski 3

4. (A) Review - Antiderivatives 12/8/2015 Calculus - Santowski 4 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 - 6e 2t

5. (B) Indefinite Integrals - Definitions 12/8/2015 Calculus - Santowski 5 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

6. (C) Review - Common Integrals 12/8/2015 Calculus - Santowski 6 Here is a list of common integrals:

7. (D) Properties of Indefinite Integrals 12/8/2015 Calculus - Santowski 7 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

8. (D) Properties of Indefinite Integrals 12/8/2015 Calculus - Santowski 8 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) = x 3 - 2x What is the answer for  f `(x)dx ….? What is the answer for d/dx  f(x)dx ….. ?

9. (E) Examples 12/8/2015 Calculus - Santowski 9  (x 4 + 3x – 9)dx =  x 4 dx + 3  xdx - 9  dx  (x 4 + 3x – 9)dx = 1/5 x 5 + 3/2 x 2 – 9x + C  e 2x dx =  sin(2x)dx =  (x 2  x)dx =  (cos  + 2sin3  )d  =  (8x + sec 2 x)dx =  (2 -  x) 2 dx =

10. (F) Examples 12/8/2015 Calculus - Santowski 10 Continue now with these questions on line Problems & Solutions with Antiderivatives / Indefinite Integrals from Visual Calculus Problems & Solutions with Antiderivatives / Indefinite Integrals from Visual Calculus

11. (G) Indefinite Integrals with Initial Conditions 12/8/2015 Calculus - Santowski 11 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  (x 3 – 3x + 1)dx if F(0) = -2 F(x) =  x 3 dx - 3  xdx +  dx = ¼x 4 – 3/2x 2 + x + C Since F(0) = -2 = ¼(0) 4 – 3/2(0) 2 + (0) + C So C = -2 and F(x) = ¼x 4 – 3/2x 2 + x - 2

12. (H) Examples – Indefinite Integrals with Initial Conditions 12/8/2015 Calculus - Santowski 12 Problems & Solutions with Antiderivatives / Indefinite Integrals and Initial Conditions from Visual Calculus Problems & Solutions with Antiderivatives / Indefinite Integrals and Initial Conditions from Visual Calculus Motion Problem #1 with Antiderivatives / Indefinite Integrals from Visual Calculus Motion Problem #1 with Antiderivatives / Indefinite Integrals from Visual Calculus Motion Problem #2 with Antiderivatives / Indefinite Integrals from Visual Calculus Motion Problem #2 with Antiderivatives / Indefinite Integrals from Visual Calculus

13. (I) Examples with Motion 12/8/2015 Calculus - Santowski 13 An object moves along a co-ordinate line with a velocity v(t) = 2 - 3t + t 2 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

14. (J) Examples – “B” Levels 12/8/2015 Calculus - Santowski 14 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 = xe x and y = xsinx

15. (K) Internet Links 12/8/2015 Calculus - Santowski 15 Calculus I (Math 2413) - Integrals from Paul Dawkins Tutorial: The Indefinite Integral from Stefan Waner's site "Everything for Calculus” Tutorial: The Indefinite Integral from Stefan Waner's site "Everything for Calculus” The Indefinite Integral from PK Ving's Problems & Solutions for Calculus 1 The Indefinite Integral from PK Ving's Problems & Solutions for Calculus 1 Karl's Calculus Tutor - Integration Using Your Rear View Mirror Karl's Calculus Tutor - Integration Using Your Rear View Mirror

16. (L) Homework 12/8/2015 Calculus - Santowski 16 Textbook, p392-394 (1) Algebra Practice: Q5-40 (AN+V) (2) Word problems: Q45-56 (economics) (3) Word problems: Q65-70 (motion)