5.b – The Substitution Rule f g x g x dx 1 Example – Optional for Pattern Learners Use WolframAlpha.com to evaluate the following. e 6 x dx. 2. Evaluate sin 4 x 12 x 3 x2 1. Evaluate 3 3. Evaluate 2 WolframAlpha Notation dx. integral[f(x),x] cos tan x sec x dx. 2 Notice that each of these are of the form f u u dx, 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? 2 The Substitution Rule – The Idea Evaluate How did you arrive at this answer? cos(e x ) e x dx without using WolframAlpha. 3 The Substitution Rule – The Idea Let’s use substitution to evaluate cos(e x ) e x dx Let u = ex and use this to complete the blanks below. f u _____________ du _________________ or du _______ __________ dx cos e x e x dx _________ ____ (in terms of u) cos e x e x dx ____________ c (in terms of x) _________ ____ __________ c (in terms of u) 4 The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f g x g x dx f u du u Since, du u g x du g x dx du g x dx In general, u will be the inside of the composition, but this isn’t always the case. 5 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. 6 Examples - Evaluate 1. 2. 4. 3. 5. 3 x x 5 dx 2 x y 3 9 3 x 2 1 2 dx 2 y 1 dy 4 6. 7. 8. sec 2 tan 2 d 9. tan 1 x dx 2 1 x 10. x sin 1 x 3 / 2 dx 1 tan sec e sin t dt 5 2 d cos t sin x dx 2 1 cos x tan x dx 7 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. 1. x dx 4 1 9x The following have extra x’s remaining in the integrand after the u substitution. This is fine. Simply solve u for x and substitute. 2. 3 x3 1 x5 dx 3. x2 dx 1 x 8