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4.5 (part 2) Integration by Substitution

Learn how to evaluate definite integrals with a change of variables, handling even and odd functions efficiently. Practice examples step-by-step.

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4.5 (part 2) Integration by Substitution

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  1. 4.5 (part 2) Integration by Substitution Greg Kelly Hanford High School Richland, Washington M.L.King Jr. Birthplace, Atlanta, GA Photo by Vickie Kelly, 2002

  2. Objectives • Use a change of variables to evaluate a definite integral. • Evaluate a definite integral involving an even or odd function.

  3. The technique is a little different for definite integrals. new limit new limit Example: We can find new limits, and then we don’t have to substitute back. We could have substituted back and used the original limits.

  4. Leave the limits out until you substitute back. This is usually more work than finding new limits Example: Using the original limits: Wrong! The limits don’t match!

  5. new limit new limit Example: Find new limits.

  6. Example: Don’t forget to use the new limits.

  7. Example:

  8. Integration of Even and Odd Functions • Even if f(-x)=f(x) • Symmetric with respect to y-axis • Odd if f(-x)=-f(x) • Symmetric with respect to origin

  9. If f is an even function then If f is an odd function then

  10. Odd

  11. In another generation or so, we might be able to use the calculator to find all integrals. Until then, remember that half the AP exam and half the nation’s college professors do not allow calculators. You must practice finding integrals by hand until you are good at it! p

  12. Homework 4.5 (page 297) #41-93 odd

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