4.5 Integration by Substitution - PowerPoint PPT Presentation

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4.5 Integration by Substitution

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Pattern recognition: Look for inside and outside functions in integral Determine what u and du would be Take integral Check by taking the derivative! – PowerPoint PPT presentation

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Title: 4.5 Integration by Substitution


1
4.5 Integration by Substitution
  • "Millions saw the apple fall, but Newton asked
    why." - Bernard Baruch

2
Objective
  • To integrate by using u-substitution

3
(No Transcript)
4
2 ways
  • Pattern recognition
  • Change in variables
  • Both use u-substitution one mentally and one
    written out

5
Pattern recognition
  • Chain rule
  • Antiderivative of chain rule

6
Antidifferentiation of a Composite Function
  • Let g be a function whose range is an interval I
    and let f be a function that is continuous on I.
    If g is differentiable on its domain and F is an
    antiderivative of f on I then

7
In other words.
  • Integral of f(u)du where u is the inside function
    and du is the derivative of the inside function

If u g(x), then du g(x)dx and
8
Recognizing patterns
9
What about
What is our constant multiple rule?
10
REMEMBER
  • The contant multiple rule only applies to
    constants!!!!!
  • You CANNOT multiply and divide by a variable and
    then move the variable outside the integral sign.
    For instance

11
In summary
  • Pattern recognition
  • Look for inside and outside functions in integral
  • Determine what u and du would be
  • Take integral
  • Check by taking the derivative!

12
Change of Variables
13
Another example
14
A third example
15
Guidelines for making a change of variables
  • 1. Choose a u g(x)
  • 2. Compute du
  • 3. Rewrite the integral in terms of u
  • 4. Evalute the integral in terms of u
  • 5. Replace u by g(x)
  • 6. Check your answer by differentiating

16
Try
17
The General Power rule for Integration
  • If g is a differentiable function of x, then
  • Equivalently, if u g(x), then

18
Change of variables for definite integrals
  • Thm If the function u g(x) has a continuous
    derivative on the closed interval a,b and f is
    continuous on the range of g, then

19
First way
20
Second way
21
Another example (way 1)
22
Way 2
23
Even and Odd functions
  • Let f be integrable on the closed interval
    -a,a
  • If f is an even function, then
  • If f is an odd function then
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