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15.2 – The Substitution Rule

This guide provides a comprehensive overview of the Substitution Rule in integration, complete with examples to illustrate the process. You will learn how to make substitutions effectively, rearranging functions to place g'(x) in front of dx. The guide emphasizes determining du/dx, multiplying by dx, and ensuring that g'(x) appears in the integrand. Following these steps will help you solve integrals where substitutions are necessary, enhancing your understanding of this essential calculus technique.

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15.2 – The Substitution Rule

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  1. 15.2 – The Substitution Rule

  2. Example 1. Evaluate 2. Evaluate 3. Evaluate

  3. The Substitution Rule u du Since,

  4. Examples - Evaluate

  5. The Substitution Rule – General Technique • Let u be g(x). You may wish to rearrange the function so that g’(x) is in front of the dx. • Determine du/dx and multiply both sides by dx. Verify that g’(x) also appears in the integrand. (It’s ok if it’s just off by a constant) • Make the substitution and everything must be written in terms of new variables u. (You need to modify the integrand first if the derivative in step 2 is off by a constant)

  6. Examples - Evaluate

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