C 7 2 indefinite integrals
Download
1 / 14

C.7.2 - Indefinite Integrals - PowerPoint PPT Presentation


  • 94 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' C.7.2 - Indefinite Integrals' - conan-sparks


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
C 7 2 indefinite integrals

C.7.2 - Indefinite Integrals

Calculus - Santowski

Calculus - Santowski


Lesson objectives
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
Fast Five

Calculus - Santowski


A review antiderivatives
(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


B indefinite integrals definitions
(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


C review common integrals
(C) Review - Common Integrals

  • Here is a list of common integrals:

Calculus - Santowski


D properties of indefinite integrals
(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


D properties of indefinite integrals1
(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)

Calculus - Santowski


E examples
(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


F examples
(F) Examples

  • Continue now with these questions on line

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

Calculus - Santowski


G indefinite integrals with initial conditions
(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


H examples indefinite integrals with initial conditions
(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


I internet links
(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

Calculus - Santowski


J homework
(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


ad