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4-5 Integration by Substitution

4-5 Integration by Substitution. Ms. Battaglia – ap calculus. Thm 4.13 Antidifferentiation of a composite function.

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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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