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5.b – The Substitution Rule

5.b – The Substitution Rule. Example – Optional for Pattern Learners. Use WolframAlpha.com to evaluate the following. 1. Evaluate. WolframAlpha Notation integral[f( x ), x ]. 2. Evaluate. 3. Evaluate.

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5.b – The Substitution Rule

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  1. 5.b – The Substitution Rule

  2. Example – Optional for Pattern Learners Use WolframAlpha.com to evaluate the following. 1. Evaluate WolframAlpha Notation integral[f(x),x] 2. Evaluate 3. Evaluate Notice that each of these are of the form where u is some function of x. If the antiderivative of f is F, what will be the answer to the indefinite integral of this form?

  3. The Substitution Rule – The Idea Evaluate without using WolframAlpha. How did you arrive at this answer?

  4. The Substitution Rule – The Idea Let’s use substitution to evaluate Let u = ex and use this to complete the blanks below.

  5. The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then In general, u will be the inside of the composition, but this isn’t always the case. u du Since,

  6. The Substitution Rule – General Technique • (Optional) Rearrange the function so that anything not in the composition is in front of the dx. • Let u be g(x) in the composition. More generally, let u be the part of the integrand such that du/dx is the other part of the integrand (with the possible exception of the coefficient of du/dx – it can be different). • Determine du/dx and multiply both sides by dx. • Divide both sides by the coefficient, if necessary. • (Only Applies to Some Integrals) If there are extra x’s remaining, solve u for x and substitute for the remaining x’s. • Perform the substitution, evaluate the integral, then perform the back substitution to get it in terms of the original variable.

  7. Examples - Evaluate

  8. Substitution Rule Twists - Examples Sometimes choosing the function for u can be challenging. Always keep in mind that you want to select a part of the integrand such that it’s derivative gives you the other part of the integrand. The following have extra x’s remaining in the integrand after the u substitution. This is fine. Simply solve u for x and substitute.

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