C.7.2 - Indefinite Integrals

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C.7.2 - Indefinite Integrals

• Calculus - Santowski

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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

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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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(C) Review - Common Integrals

• Here is a list of common integrals

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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
  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 .. ?

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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
• ?(x2?x)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) 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

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(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

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(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

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(L) Homework

• Textbook, p392-394
• (1) Algebra Practice Q5-40 (ANV)
• (2) Word problems Q45-56 (economics)
• (3) Word problems Q65-70 (motion)