4 2 Antiderivatives Substitution 4 2 Antiderivatives Substitution i Watch videos 46 and 47 for Section 4 2 Complete the rst two pages of these notes while watching the videos This needs to be done before coming to class In the previous section we learned introductory techniques for nding antiderivatives In this section we will learn a more advanced technique to nd the antiderivative of a function that is a composition of functions cid 17 3 cid 19 cid 18 cid 16 cid 17 2 4x 12x d dx 2x2 4 cid 16 cid 16 cid 17 2 4x 12x cid 16 cid 17 2 how do we nd F 2x2 4 cid 16 In Chapter 2 we found d dx The process of reversing the Chain Rule to nd an antiderivative is called substitution or u substitution F x 3 2x2 4 2x2 4 3 2x2 4 cid 17 2 So now the question is given The Substitution Method Recall the Chain Rule enables us to nd the derivative of a function that is a composition of functions The method of substitution works by reversing the Chain Rule In other words cid 2 f g x cid 3 f cid 48 g x g cid 48 x f cid 48 g x g cid 48 x dx f g x C The idea behind substitution is that if part of the integrand contains a composite function that was found as a result of using the Chain Rule we identify the inside function of the composite function and its corresponding derivative and perform a substitution so that we can obtain a new integral in which we can use the Introductory Antiderivative Rules to nd the antiderivative In other words we identify what we referred to as g x and g cid 48 x when learning the Chain Rule so we can obtain an integral of the form cid 90 f cid 48 u du where u g x and du g cid 48 x dx Let s return to the question above Given F x 3 2x2 4 2x2 4 cid 16 cid 17 2 4x 12x cid 16 cid 17 2 how do we nd F d dx cid 90 d dx N When we write du g cid 48 x dx we are treating du and dx as the change in u and x respectively du and dx are called di erentials Any further discussion about the di erentials du and dx is beyond the scope of this course so when nding the derivative of u g x we will simply write du g cid 48 x dx TAMU Companion to Calculus for Business Social Sciences by Allen A and Orchard P 2021 1 4 2 Antiderivatives Substitution To perform the method of substitution we 1 Let u g x 2 Write du g cid 48 x dx 3 Rewrite the integral in terms of u and du cid 4 Example 1 Find each of the following inde nite integrals a 3x2 x3 7 8 dx 4 Integrate i e nd the antiderivative 5 Substitute back for u cid 90 cid 90 cid 90 b 21x2e7x3 2 dx c 10x4 2x5 10 dx TAMU Companion to Calculus for Business Social Sciences by Allen A and Orchard P 2021 cid 4 2 4 2 Antiderivatives Substitution The integrands in Example 1 were selected for a speci c purpose to demonstrate the three possible cases we may encounter when using the method of substitution to nd an antiderivative We summarize these three cases below When using the method of substitution to nd an antiderivative looking for one of the following formats in the integrand may help you determine the correct substitution u g x where g x is the inside function Possible Substitution Cases cid 90 cid 90 cid 90 1 1 2 3 g cid 48 x dx g x g x n g cid 48 x dx where n is any real number with n cid 44 1 b g x g cid 48 x dx where b is any positive real number N There may be more than one choice for u that will allow us to nd the antiderivative of a function using the method of substitution If we can rewrite the given integral entirely in terms of u and du and nd the resulting antiderivative using the Introductory Antiderivative Rules then we have chosen a substitution that will work If not we must go back and select a di erent u cid 4 Example 2 Find each of the following inde nite integrals cid 90 10x4 9x3 3 8x5 9x4 dx a TAMU Companion to Calculus for Business Social Sciences by Allen A and Orchard P 2021 3 cid 90 4 2 Antiderivatives Substitution b 8x 12 107x2 21x dx cid 90 c e3x e3x 5 dx TAMU Companion to Calculus for Business Social Sciences by Allen A and Orchard P 2021 cid 4 4 4 2 Antiderivatives Substitution We summarize this process formally below The Substitution Method To nd the antiderivative of a function using the method of substitution follow these ve steps 1 Look carefully at the integrand and select u g x so that g cid 48 x is also part of the integrand or a multiple of it Remember when selecting u g x that g x is the inside function of the composite function in the integrand 2 Take the derivative of u g x to obtain du g cid 48 x dx 3 Rewrite the integral in terms of u and du only 4 Find N If you cannot fully rewrite the integral without the original variable then in most cases the choice of u will not work In this case you must return to step 1 and try selecting a di erent u f cid 48 u du using the Introductory Antiderivative Rules cid 82 N We may need to assess our choice of u at this step If we cannot nd an antiderivative of f cid 48 u using the Introductory Antiderivative Rules then our choice of u will not work We must return to step 1 and choose a di erent part of the integrand for u 5 Substitute u g x so the antiderivative is in terms of the original variable x Specific Antiderivatives At the heart of it nding speci c antiderivatives with the method of substitution is the same as nding them without the method of substitution which we learned in Section 4 1 Find the most general antiderivative and then use the point given to nd the value of the constant of integration C cid 4 Example 3 Find y if y cid 48 and y e 10 ln x 5 x TAMU Companion to Calculus for Business Social Sciences by Allen A and Orchard P 2021 cid 4 5 4 2 Antiderivatives Substitution Applications We can use the method of substitution to solve problems involving the same types of applications we encountered using introductory antidi erentiation techniques only now we can apply the method of substitution to more complicated functions cid 4 Example 4 The marginal cost function for Feynman s …
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