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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.