Use the log rule to integrate a rational function
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
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.
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?
Ex 4 p. 333 Recognizing Quotient Forms of the Log Rule
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
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!
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?”
Sorry, # 4 is not available. So memorize, memorize, memorize and be creative!
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.