1. 4.1  2012 Pearson Education, Inc. All rights reserved Slide 4.1-1 Antidifferentiation OBJECTIVE Find an antiderivative of a function. Evaluate indefinite integrals using the basic integration formulas. Use initial conditions, or boundary conditions, to determine an antiderivative.

2.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-2 THEOREM 1 The antiderivative of is the set of functions such that The constant C is called the constant of integration. 4.1 Antidifferentiation

3.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-3 Integrals and Integration Antidifferentiating is often called integration. To indicate the antiderivative of x 2 is x 3 /3 +C, we write where the notation is used to represent the antiderivative of f (x). More generally, where F(x) + C is the general form of the antiderivative of f (x). 4.1 Antidifferentiation

4.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-4 Example 1: Determine these indefinite integrals. That is, find the antiderivative of each integrand: a.) b.) c.) d.) 4.1 Antidifferentiation

5.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-5 THEOREM 2: Basic Integration Formulas 4.1 Antidifferentiation

6.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-6 THEOREM 2: Basic Integration Formulas (continued) 4.1 Antidifferentiation

7.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-7 Example 2: Use the Power Rule of Antidifferentiation to determine these indefinite integrals: a.) b.) 4.1 Antidifferentiation

8.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-8 Example 2 (Continued) c) We note that Therefore 4.1 Antidifferentiation

9.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-9 Example 2 (Concluded) d) We note that Therefore 4.1 Antidifferentiation

10.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-10 4.1 Antidifferentiation Quick Check 1 Determine these indefinite integrals: a.) b.) c.) d.)

11.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-11 Example 3: Determine the indefinite integral Since we know that it is reasonable to make this initial guess: But this is (slightly) wrong, since 4.1 Antidifferentiation

12.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-12 Example 3(Concluded): We modify our guess by inserting to obtain the correct antiderivative: This checks: 4.1 Antidifferentiation

13.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-13 4.1 Antidifferentiation Quick Check 2 Find each antiderivative: a.) b.)

14.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-14 THEOREM 3 Properties of Antidifferentiation (The integral of a constant times a function is the constant times the integral of the function.) (The integral of a sum or difference is the sum or difference of the integrals.) 4.1 Antidifferentiation

15.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-15 Example 4: Determine these indefinite integrals. Assume x > 0. a.) We antidifferentiate each term separately: 4.1 Antidifferentiation

16.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-16 Example 4 (Concluded): b) We algebraically simplify the integrand by noting that x is a common denominator and then reducing each ratio as much as possible: Therefore, 4.1 Antidifferentiation

17.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-17 4.1 Antidifferentiation Quick Check 3 Determine these indefinite integrals: a.) b.) c.)

18.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-18 Example 5: Find the function f such that First find f (x) by integrating. 4.1 Antidifferentiation

19.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-19 Example 5 (concluded): Then, the initial condition allows us to find C. Thus, 4.1 Antidifferentiation

20.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-20 4.1 Antidifferentiation Section Summary The antiderivative of a function is a set of functions F(x) such that where the constant C is called the constant of integration. An antiderivative is denoted by an indefinite integral using the integral sign, If is an antiderivative of we write We check the correctness of an antiderivative we have found by differentiating it.

21.  2012 Pearson Education, Inc. All rights reserved Slide 4.1-21 4.1 Antidifferentiation Section Summary Continued The Constant Rule of Antidifferentiation is The Power Rule of Antidifferentiation is The Natural Logarithm Rule of Antidifferentiation is The Exponential Rule (base e) of Antidifferentiation is An initial condition is an ordered pair that is a solution of a particular antiderivative of an integrand.