# 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 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)

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