4.5 Integration by Substitution

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# 4.5 Integration by Substitution - PowerPoint PPT Presentation

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.

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## 4.5 Integration by Substitution

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### 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
• Chain rule:
• Antiderivative of chain rule:
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….
• 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 is our constant multiple rule?

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…
• 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
• 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)