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!

'ln

udx u Cu

Since du=u’dx, another form is

Ex 1 p. 332 Using the log rule for integration

5dxx

15 dxx

4dxx

14 dxx

4ln x C 4ln x C

5ln x C 5

ln x C

Ex 2 p. 332 Using the Log Rule with a change of variables

Find1

6 1dx

x Let u = 6x – 1. Then du = 6dx so I need a 6 multiplied into the integral and 1/6 on the outside

1 16

6 6 1dx

x

Substitute in u and du

1 1

6duu

1

ln6

u C

Apply Log Rule

Back-substitute

1ln 6 1

6x C

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 ofthe x-axis and the line x = 3

2

4

1

xy

x

3

20

4

1

xdx

x Let u = x2 + 1. Then du = (2x)dx and rewrite to have du in numerator.

3

20

22

1

xdx

x

32

02ln( 1)x 2(ln10 ln1) Why didn’t I need

absolute value in log?10

2(ln ) 2(ln10)1

4.605

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

2

3

3 1.

xa dx

x x

3 2, 3 1u x x du x

3ln +Cx x

2sec.

tan

xb dx

x2tan , secu x du xdx ln tan +C x

2

3

1.

3

xc dx

x x

3 2 23 , (3 3) 3( 1)u x x du x dx x dx 2

3

1 3( 1)

3 3

xdx

x x

31 ln 33 x x C

1.

4 5d dx

x 4 5, 4u x du dx 1 4

=4 4 5

dxx

1 ln 4 54 x C

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

22 7 3

2

x xdx

x

2

2

2 112 2 7 3

2 4

11 3

11 22

19

xx x x

x x

x

x

19

2 11 2

x dxx

Let u = x – 2. Then du = dx

2 11 19ln 2x x x C

Ex 6 p. 334 Change of Variables with the Log Rule (in disguise!)

3

( 2)

1

x xdx

x

let 1. Then , 1u x du dx x u

3

1 1 2u udu

u

With rewrite in terms of u

2

3

1udu

u

3

1 1du duu u

2

ln2

uu C

2

1ln

2u C

u

Back-substitute

2

1ln 1

2 1x C

x

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 2nd 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.