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C.2.1 – Integration by Substitution

C.2.1 – Integration by Substitution. Calculus - Santowski. Lesson Objectives. Use the method of substitution to integrate simple composite power, exponential, logarithmic and trigonometric functions both in a mathematical context and in a real world problem context. Fast Five.

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C.2.1 – Integration by Substitution

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  1. C.2.1 – Integration by Substitution Calculus - Santowski Calculus - Santowski

  2. Lesson Objectives • Use the method of substitution to integrate simple composite power, exponential, logarithmic and trigonometric functions both in a mathematical context and in a real world problem context Calculus - Santowski

  3. Fast Five • Differentiate the following functions: Calculus - Santowski

  4. (A) Introduction • At this point, we know how to do simple integrals wherein we simply apply our standard integral “formulas” • But, similar to our investigation into differential calculus, functions become more difficult/challenging, so we developed new “rules” for working with more complex functions • Likewise, we will see the same idea in integral calculus and we shall introduce 2 methods that will help us to work with integrals Calculus - Santowski

  5. (B) “Simple” Examples ???? • Find the following: • Now, try these: Calculus - Santowski

  6. (C) Looking for Patterns • Alright, let’s use the TI-89 to help us with some of the following integrals: • Now, look at our fast 5 • Now, let’s look for patterns??? • Examples Calculus - Santowski

  7. (C) Looking for Patterns • So, in all the integrals presented here, we see that some part of the function to be integrated is a COMPOSED function and then the second pattern we observe is that we also see some of the derivative of the “inner” function appearing in the integral • Here are more examples to illustrate our “pattern” Calculus - Santowski

  8. (D) Generalization from our Pattern • So we can make the following generalization from our observation of patterns: • But the question becomes: how do we know what substitution to make??? • Generalization: ask yourself what portion of the integrand has an inside function and can you do the integral with that inside function present. If you can’t then there is a pretty good chance that the inside function will be the substitution. Calculus - Santowski

  9. (E) Working out Some Examples • Integrate Calculus - Santowski

  10. (E) Working out Some Examples • Integrate Calculus - Santowski

  11. (F) Further Examples • Integrate the following: Calculus - Santowski

  12. (G) Homework • Stewart text, 1998, §5.5, p400 – 402 • (1) indefinite integrals from Q1-32  ANV • (2) Q33,34,35 • (3) definite integrals  Q37,41,43,45,51 • (4) Word Problems  Q64-66 • DAY 2 • (4) Q53-58,61-63,67,68,70 Calculus - Santowski

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