208 CHAPTER 8. METHODS FOR FINDING ANTIDERIVATIVES

4. Find the constants, ci, pi, and qi. Use whatever method you like. You might seeif you can make up new ways to do this if you like. If you have followed steps 1- 3 correctly, it will work out. However, be sure to search for something which isactually there. Otherwise, you won’t find it.

The above technique for finding the coefficients is fine but some people like to do itother ways. It really does not matter how you do it. Here is another example.

Example 8.7.9 Find the partial fractions expansion for

15x4 +44x3 +71x2 +64x+28+2x5

(x+2)2 (x2 +2x+2)2

The degree of the top is 4 and the degree of the bottom is 6 so you do not need to dolong division. You do have to look for the right thing however. The correct form for thepartial fractions expansion is

ax+2

+b

(x+2)2 +cx+d

x2 +2x+2+

ex+ f

(x2 +2x+2)2

=15x4 +44x3 +71x2 +64x+28+2x5

(x+2)2 (x2 +2x+2)2

Multiply both sides by (x+2)2 and then plug in x =−2.

b =15(−2)4 +44(−2)3 +71(−2)2 +64(−2)+28+2(−2)5(

(−2)2 +2(−2)+2)2 = 2

Now subtract the term involving b from both sides.

ax+2

+cx+d

x2 +2x+2+

ex+ f

(x2 +2x+2)2 =

15x4 +44x3 +71x2 +64x+28+2x5

(x+2)2 (x2 +2x+2)2 − 2

(x+2)2

=1

(x+2)(x2 +2x+2)2

(2x4 +9x3 +18x2 +19x+10

)Multiply both sides by x+2 and plug in x =−2.

a =1(

(−2)2 +2(−2)+2)2

(2(−2)4 +9(−2)3 +18(−2)2 +19(−2)+10

)= 1

Subtract this term involving a from both sides.

cx+dx2 +2x+2

+ex+ f

(x2 +2x+2)2