4.1 ANTIDERIVATIVES An/deriva/ves We need to find a way to undo differen/a/on. That is, given a func/on, f , we’d like to find another func/on F such that F’(x) = f (x). We call such a func/on F an an#deriva#ve of f. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 3 4.1 ANTIDERIVATIVES EXAMPLE 1.1 Finding Several An/deriva/ves of a Given Func/on © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 4 4.1 ANTIDERIVATIVES EXAMPLE 1.1 Finding Several An/deriva/ves of a Given Func/on Solu/on © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 5 4.1 ANTIDERIVATIVES The An/deriva/ves of f Observe that if F is any an/deriva/ve of f and c is any constant, then Are there any other an/deriva/ves of f (x) besides F(x) + c? The answer, as provided in Theorem 1.1, is no. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 6 4.1 ANTIDERIVATIVES THEOREM 1.1 Suppose that F and G are both an/deriva/ves of f on an interval I . Then, G(x) = F(x) + c, for some constant c. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 7 4.1 ANTIDERIVATIVES PROOF Since F and G are both an/deriva/ves for f, we have that G’(x) = F’(x). It now follows, from Corollary 10.1 in sec/on 2.10, that G(x) = F(x) + c, for some constant c, as desired. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 8 4.1 ANTIDERIVATIVES DEFINITION 1.1 Let F be any an/deriva/ve of f on an interval I . The indefinite integral of f (x) (with respect to x) on I is defined by where c is an arbitrary constant (the constant of integra#on). © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 9 4.1 ANTIDERIVATIVES Integra/on The process of compu/ng an integral is called integra#on. Here, f (x) is called the integrand and the term dx iden/fies x as the variable of integra#on. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 10 4.1 ANTIDERIVATIVES EXAMPLE 1.2 An Indefinite Integral © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 11 4.1 ANTIDERIVATIVES EXAMPLE 1.2 An Indefinite Integral Solu/on You should recognize 3x2 as the deriva/ve of x3 and so, © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 12 4.1 ANTIDERIVATIVES EXAMPLE 1.3 Evalua/ng an Indefinite Integral © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 13 4.1 ANTIDERIVATIVES EXAMPLE 1.3 Evalua/ng an Indefinite Integral Solu/on © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 14 4.1 ANTIDERIVATIVES Deriva/ve Rules Give Integra/on Rules Every differen/a/on rule gives rise to a corresponding integra/on rule. For instance, recall that for every ra/onal power, r, Likewise, This proves the following result. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 15 4.1 ANTIDERIVATIVES THEOREM 1.2 For any ra/onal power r ≠ −1, Here, if r < −1, the interval I on which this is defined can be any interval that does not include x = 0. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 16 4.1 ANTIDERIVATIVES EXAMPLE 1.4 Using the Power Rule © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 17 4.1 ANTIDERIVATIVES EXAMPLE 1.4 Using the Power Rule Solu/on From the power rule, © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 18 4.1 ANTIDERIVATIVES EXAMPLE 1.5 The Power Rule with a Nega/ve Exponent © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 19 4.1 ANTIDERIVATIVES EXAMPLE 1.5 The Power Rule with a Nega/ve Exponent Solu/on In any interval not containing 0, © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 20 4.1 ANTIDERIVATIVES EXAMPLE 1.6 The Power Rule with a Frac/onal Exponent © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 21 4.1 ANTIDERIVATIVES EXAMPLE 1.6 The Power Rule with a Frac/onal Exponent Solu/on (a) (b) © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 22 4.1 ANTIDERIVATIVES Some Integra/on Rules By reversing any deriva/ve formula, we get a corresponding integra/on formula. The formulas here can be obtained using this strategy. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 23 4.1 ANTIDERIVATIVES Some Integra/on Rules © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 24 4.1 ANTIDERIVATIVES THEOREM 1.3 Suppose that f (x) and g(x) have an/deriva/ves. Then, for any constants, a and b, Note that Theorem 1.3 says that we can easily compute integrals of sums, differences and constant mul/ples of func/ons. © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 25 4.1 ANTIDERIVATIVES PROOF We have that It then follows that © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 26 4.1 ANTIDERIVATIVES EXAMPLE 1.7 An Indefinite Integral of a Sum © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 27 4.1 ANTIDERIVATIVES EXAMPLE 1.7 An Indefinite Integral of a Sum Solu/on © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 28 4.1 ANTIDERIVATIVES EXAMPLE 1.8 An Indefinite Integral of a Difference © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 29 4.1 ANTIDERIVATIVES EXAMPLE 1.8 An Indefinite Integral of a Difference Solu/on © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 30 4.1 ANTIDERIVATIVES Differen/a/ng ln|x| We recall that for x > 0, Now consider ln|x| for x > 0. In this case ln|x| = ln x, so that For x < 0, ln|x| = ln (−x), so that © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 31 4.1 ANTIDERIVATIVES THEOREM 1.4 © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 32 4.1 ANTIDERIVATIVES COROLLARY 1.1 & 1.2 1.1 1.2 © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 33 4.1 ANTIDERIVATIVES EXAMPLE 1.10 The Indefinite Integral of a Frac/on of the Form f ’(x)/f(x) © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 34 4.1 ANTIDERIVATIVES EXAMPLE 1.10 The Indefinite Integral of a Frac/on of the Form f ’(x)/f(x) Solu/on © The McGraw-­‐Hill Companies, Inc. Permission required for reproduc/on or display. Slide 35