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Warm-up:

Warm-up:. Evaluate the integrals. 1) 2). Warm-up:. Evaluate the integrals. 1) 2). Warm-up:. Evaluate the integrals. 1) 2). Integration by Substitution. Section 6.3. U-Substitution. Can be used to transform complicated integration problems into simpler ones. U-Substitution.

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Warm-up:

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  1. Warm-up: Evaluate the integrals. 1) 2)

  2. Warm-up: Evaluate the integrals. 1) 2)

  3. Warm-up: Evaluate the integrals. 1) 2)

  4. Integration by Substitution Section 6.3

  5. U-Substitution • Can be used to transform complicated integration problems into simpler ones.

  6. U-Substitution • Can be used to transform complicated integration problems into simpler ones. * In general, there are no hard and fast rules for choosing u, and in some problems no choice of u will work. For those problems, we have other methods that will be discussed later. Choosing a u that is appropriate comes with practice and experience, but the guidelines will help to gain a better understanding.

  7. Guidelines for Choosing U • Look for a substitution that would produce an integral expressed entirely in terms of u and du. • Evaluate the integral in terms of u. • Replace the u, so that your final answer is in terms of x.

  8. Examples • Evaluate • Evaluate

  9. Examples • Evaluate • Evaluate

  10. More Examples • Evaluate • Evaluate

  11. More Examples • Evaluate • Evaluate

  12. More difficult Examples • Evaluate • Evaluate

  13. More difficult Examples • Evaluate • Evaluate

  14. More difficult Examples • Evaluate • Evaluate

  15. More difficult Examples • Evaluate • Evaluate

  16. Even More Difficult Examples • Evaluate • Evaluate

  17. Even More Difficult Examples • Evaluate • Evaluate

  18. Even More Difficult Examples • Evaluate • Evaluate

  19. Difficult Examples (Conclusion) • Evaluate • Evaluate

  20. Difficult Examples (Conclusion) • Evaluate • Evaluate

  21. Homework: page 371 # 1 – 23 odd

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