8.7. PARTIAL FRACTIONS 207

The reason there are two terms devoted to (x+2) is that this is squared. Computing theconstants yields

3x+7

(x+2)2 (x+3)=

1

(x+2)2 +2

x+2− 2

x+3

and therefore,∫ 3x+7

(x+2)2 (x+3)dx =− 1

x+2+2ln |x+2|−2ln |x+3|+C.

Example 8.7.8 Find the proper form for the partial fractions expansion of

x3 +7x+9

(x2 +2x+2)3 (x+2)2 (x+1)(x2 +1).

First check to see if the degree of the numerator is smaller than the degree of the de-nominator. Since this is the case, look for a partial fractions decomposition in the followingform.

ax+b(x2 +2x+2)

+cx+d

(x2 +2x+2)2 +ex+ f

(x2 +2x+2)3+

A(x+2)

+B

(x+2)2 +D

(x+1)+

gx+hx2 +1

.

These examples illustrate what to do when using the method of partial fractions. You firstcheck to be sure the degree of the numerator is less than the degree of the denominator. Ifthis is not so, do a long division. Then you factor the denominator into a product of factors,some linear of the form ax+b and others quadratic, ax2 +bx+ c which cannot be factoredfurther. Next follow the procedure illustrated in the above examples and summarized below.

Warning: When you use partial fractions, be sure you look for something which isof the right form. Otherwise you may be looking for something which is not there. Therules are summarized next.

Rules For Finding Partial Fractions Expansion Of A Rational Function

1. Check to see if the numerator has smaller degree than the denominator. If this is notso, correct the situation by doing long division.

2. Factor the denominator into a product of linear factors, (Things like (ax+b)) andirreducible quadratic factors, (Things like

(ax2 +bx+ c

)where b2 −4ac < 0.)1

3. Let m,n be positive integers. Corresponding to (ax+b)m in the denominator, youshould have a sum of the form ∑

mi=1

ci(ax+b)i in the partial fractions expansion. Here the

ci are the constants to be found. Corresponding to(ax2 +bx+ c

)n in the denominatorwhere b2−4ac< 0, you should have a sum of the form ∑

mi=1

pix+qi

(ax2+bx+c)i in the partial

fractions expansion. Here the pi and qi are to be found.

1Of course this factoring of the denominator is easier said than done. In general you cannot do it at all. Ofcourse there are big theorems which guarantee the existence of such a factorization but these theorems do not tellhow to find it. This is an example of the gap between theory and practice which permeates mathematics.