Chapter 7 – Techniques of Integration

1. Chapter 7 – Techniques of Integration 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

2. When do we use it? • Differentiation is an easier straightforward process as opposed to Integration. • Integration is a more challenging process. • When integrating we will have to use algebra manipulation, substitution, integration by parts , partial fractions, and many times all of the above. • It is important that you memorize the Table of Integration Formulas on page 495 in your book. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

3. Strategy for integration • Step 1: Simplify the integrand if possible. • Step 2: Look for an obvious substitution. • Step 3: Classify the Integrand according to its form • Step 4: Try again (substitution, parts, manipulations) The following slides will illustrate some examples of these steps. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

4. Step 1:Simplify the integrand • Use algebraic manipulations to simplify the integrand Examples 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

5. Step 2: Use Substitution Try to find a function whose derivative occurs in the integrand. • Examples 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

6. Step 3: Classify and Integral by Its Form • Trig functions - If our function is a product of powers of the trig functions then use trig substitution. • Rational functions - If our function is a rational functions then use partial fractions. • Integration by parts - If our function is a product of a power of x(or a polynomial) and a transcendental function then use integration by parts. • Radicals– If our function is a radical, we have certain options • If occurs we use trig substitution • If occurs we use the rationalizing substitution 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

7. Step 4: Try Again • If the first three steps have not produced the answer, remember that there are basically only two methods of integration: substitution and parts. • Try substitution. Even if no substitution is obvious (Step 2), try again. • Try parts. Although integration by parts is used most of the time on products (step 3), we can use it on single functions that are inverse functions. • Manipulate the integrand. Try rationalizing the denominator or trig identities. • Relate the problem to previous problems. • Use several methods. Sometimes two or three methods are needed to evaluate an integral. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

8. Can we integrate all continuous functions? • No. There are some functions that we can’t integrate in terms of functions we know. We will learn in chapter 11 how to express these functions as an infinite series. Some integrals we can’t evaluate. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

9. Examples – Page 499 • Evaluate the integral. • NIB. • 14. • NIB. • 20. • 34. • 44. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

10. Examples – Page 500 • Evaluate the integral. • NIB. • 75. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers

11. Examples – Page 504 Use the table of integrals on Reference Pages 6-10 to evaluate the integral. State the formula you used. NIB. 11. 12. 24. 26. 30. 7.5 Strategy for Integration 7.6 Integration Using Tables and Computers