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4-5 INTEGRATION BY SUBSTITUTION
MS. BATTAGLIA – AP CALCULUS
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THM 4.13 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
Letting u = g(x) gives du = g’(x)dx and
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EXPLORATION
The integrand in each of the following integrals fits the pattern f(g(x))g’(x). Identify the pattern and use the result to evaluate the integral.
The next three integrals are similar to the first three. Show how you can multiply and divide by a constant to evaluate these integrals.
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RECOGNIZING THE F(G(X))G’(X) PATTERN
Find
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MULTIPLYING AND DIVIDING BY A CONSTANT
Find
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CHANGE IN VARIABLES
If u=g(x) then du=g’(x)dx and the integral takes the form
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CHANGE OF VARIABLES
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CHANGE OF VARIABLES
Find
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CHANGE OF VARIABLES
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GUIDELINES FOR MAKING A CHANGE OF VARIABLES
1. Choose a substitution u=g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power.
2. Compute du=g’(x)dx
3. Rewrite the integral in terms of the variable u.
4. Finding the resulting integral in terms of u.
5. Replace u by g(x) to obtain an antiderivative in terms of x.
6. Check your answer by differentiating.
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THE GENERAL POWER RULE FOR INTEGRATION
If g is a differentiable function of x, then
Equivalently, if u=g(x), then
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SUBSTITUTION AND THE GENERAL POWER RULE
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SUBSTITUTION AND THE GENERAL POWER RULE
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CHANGE OF VARIABLES FOR DEFINITE INTEGRALS
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
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CHANGE OF VARIABLES
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CHANGE OF VARIABLES
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INTEGRATION OF EVEN AND ODD FUNCTIONS
Let f be integrable on the closed interval [-a,a]
1. If f is an even function, then
2. If f is an odd function, then
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INTEGRATION OF AN ODD FUNCTION
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HOMEWORK
Read 4.5 Page 306
#2, 3, 7-10, 13, 29, 31, 33, 37, 39, 47, 53, 107