1 5.b – The Substitution Rule

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

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 = e x 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 Since, u du In general, u will be the inside of the composition, but this isn’t always the case.

6 The Substitution Rule – General Technique 1.(Optional) Rearrange the function so that anything not in the composition is in front of the dx. 2.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). 3.Determine du/dx and multiply both sides by dx. 4.Divide both sides by the coefficient, if necessary. 5.(Only Applies to Some Integrals) If there are extra x’s remaining, solve u for x and substitute for the remaining x’s. 6.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.