The document discusses integration rules and techniques for rational functions using the log rule. It provides examples of using the log rule with substitutions, recognizing quotients in disguised forms, using long division to reveal quotients, and changing variables. It emphasizes mastering the "form-fitting" nature of integration to apply the appropriate rules compared to differentiation which is more straightforward. Students are encouraged to memorize the integration and derivative rules.

1. Use the log rule to integrate a rational function

2. The differentiation rules you learned in 5.1 lead to the integration rules for 5.2. Remember, u is a function of x, and you must have the chain du in the integral to unchain as you integrate! Since du=u’dx, another form is

3. Ex 1 p. 332 Using the log rule for integration

4. Ex 2 p. 332 Using the Log Rule with a change of variables Find Let u = 6x – 1. Then du = 6dx so I need a 6 multiplied into the integral and 1/6 on the outside Substitute in u and du Apply Log Rule Back-substitute Write down u and du even if you don’t do the integration with a substitution! It helps.

5. In the next example, using the alternative form of the Log Rule helps. Look for quotients in which the numerator is the derivative of the denominator. Ex 3 p. 333 Finding area with the Log Rule Find the area bounded by the graph of the x-axis and the line x = 3 Let u = x 2 + 1. Then du = (2x)dx and rewrite to have du in numerator. Why didn’t I need absolute value in log?

6. Ex 4 p. 333 Recognizing Quotient Forms of the Log Rule

7. Sometimes integrals that the log rule works for come in disguise. For example, if the numerator has a degree that is greater than or equal to the denominator, long division might reveal a form that works. Ex 5 p. 334 Using Long Division before Integrating Let u = x – 2. Then du = dx

8. Ex 6 p. 334 Change of Variables with the Log Rule (in disguise!) With rewrite in terms of u Back-substitute Remember, can only split up if single term denominator!

9. Example 5 and 6 use methods involving rewriting a disguised integrand so that it fits one or more of the basic integration formulas. To become a pro, you must master the “form-fitting” nature of integration. Derivatives are very straight-forward. “ Here is the question; what is the answer?” Integration is more like “ Here is the answer; what is the question?”

10. Sorry, # 4 is not available. So memorize, memorize, memorize and be creative!

11. 5.2a p. 338 1-25 every other odd, 45, 61, 63, 67, 71, 91, 93 A powerpoint with integration included is on my website under 2 nd trimester. We might not have learned all the rules yet, but get a head-start on memorization by downloading it and practicing until you know all derivative and integration rules by heart.