tailieunhanh - Calculus and its applications: 4.5

"Calculus and its applications: " Calculus and its applications - integration techniques-substitution have objective: evaluate integrals using substitution. | 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution OBJECTIVE Evaluate integrals using substitution. 2012 Pearson Education, Inc. All rights reserved Slide The following formulas provide a basis for an integration technique called substitution. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 1: For y = f (x) = x3, find dy. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 2: For u = F(x) = x2/3, find du. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 3: For find dy. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution Quick Check 1 Find each differential. a.) b.) c.) d.) 2012 Pearson Education, Inc. All rights reserved Slide Example 4: Evaluate: Note that 3x2 is the derivative of x3. Thus, Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution Quick Check 2 Evaluate: 2012 Pearson Education, Inc. All rights reserved Slide Example 5: Evaluate: Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 6: Evaluate: Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution Quick Check 3 Evaluate: 2012 Pearson Education, Inc. All rights reserved Slide Example 7: Evaluate: Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 8: Evaluate: Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution Quick Check 4 Evaluate: 2012 Pearson Education, Inc. All rights reserved Slide Example 9: Evaluate: Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 10: Evaluate: We first find the indefinite integral and then evaluate the integral over [0, 1]. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 10 (concluded): Then, we have Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution Section Summary Integration by substitution is the reverse of applying the Chain Rule of Differentiation. The substitution is reversed after the integration has been performed. Results should be checked using differentiation. | 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution OBJECTIVE Evaluate integrals using substitution. 2012 Pearson Education, Inc. All rights reserved Slide The following formulas provide a basis for an integration technique called substitution. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 1: For y = f (x) = x3, find dy. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 2: For u = F(x) = x2/3, find du. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Example 3: For find dy. Integration Techniques: Substitution 2012 Pearson Education, Inc. All rights reserved Slide Integration Techniques: Substitution Quick Check 1 Find each differential. a.) b.) c.) d.) 2012 Pearson Education, Inc. All rights reserved Slide Example 4: .