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4-5 Integration by Substitution

Explore integration by substitution using the fundamental theorem, change of variables, and general power rule to simplify and evaluate composite integrals efficiently. Practice with examples and guidelines for successful substitutions.

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4-5 Integration by Substitution

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  1. 4-5 Integration by Substitution Ms. Battaglia – ap calculus

  2. 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

  3. 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.

  4. Recognizing the f(g(x))g’(x) pattern • Find

  5. Multiplying and dividing by a constant • Find

  6. Change in variables • If u=g(x) then du=g’(x)dx and the integral takes the form

  7. Change of variables

  8. Change of variables • Find

  9. Change of variables

  10. Guidelines for making a change of variables • 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. • Compute du=g’(x)dx • Rewrite the integral in terms of the variable u. • Finding the resulting integral in terms of u. • Replace u by g(x) to obtain an antiderivative in terms of x. • Check your answer by differentiating.

  11. The general power rule for integration • If g is a differentiable function of x, then • Equivalently, if u=g(x), then

  12. Substitution and the general power rule

  13. Substitution and the general power rule

  14. 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

  15. Change of variables

  16. Change of variables

  17. Integration of 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

  18. Integration of an odd function

  19. Homework • Read 4.5 Page 306 • #2, 3, 7-10, 13, 29, 31, 33, 37, 39, 47, 53, 107

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