4 5 integration by substitution n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
4.5 Integration by Substitution PowerPoint Presentation
Download Presentation
4.5 Integration by Substitution

Loading in 2 Seconds...

play fullscreen
1 / 23

4.5 Integration by Substitution - PowerPoint PPT Presentation


  • 85 Views
  • Uploaded on

4.5 Integration by Substitution. "Millions saw the apple fall, but Newton asked why." -– Bernard Baruch . Objective. To integrate by using u-substitution. 2 ways…. Pattern recognition Change in variables Both use u-substitution… one mentally and one written out. Pattern recognition.

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

4.5 Integration by Substitution


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
4 5 integration by substitution

4.5 Integration by Substitution

"Millions saw the apple fall, but Newton asked why." -– Bernard Baruch

objective
Objective
  • To integrate by using u-substitution
2 ways
2 ways…
  • Pattern recognition
  • Change in variables
  • Both use u-substitution… one mentally and one written out
pattern recognition
Pattern recognition
  • Chain rule:
  • Antiderivative of chain rule:
antidifferentiation of a composite function
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…
in other words
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

what about
What about…

What is our constant multiple rule?

remember
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…
in summary
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!
guidelines for making a change of variables
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
the general power rule for integration
The General Power rule for Integration
  • If g is a differentiable function of x, then
  • Equivalently, if u = g(x), then
change of variables for definite integrals
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
even and odd functions
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