Example 41 __
BC ONLY†C. INTEGRATION BY PARTIAL FRACTIONS
The method of partial fractions makes it possible to express a rational function
as a sum of simpler fractions. Here f (x) and g(x) are real polynomials in x
and it is assumed that is a proper fraction; that is, that f (x) is of lower degree
than g(x). If not, we divide f (x) by g(x) to express the given rational function as
the sum of a polynomial and a proper rational function. Thus,
,where the fraction on the right is proper.
Theoretically, every real polynomial can be expressed as a product of (powers
of) real linear factors and (powers of) real quadratic factors.†
In the following, the capital letters denote constants to be determined. We
consider only nonrepeating linear factors. For each distinct linear factor (x − a)
of g(x) we set up one partial fraction of the type . The techniques for
determining the unknown constants are illustrated in the following examples.
BC ONLYExample 42 __
Find .
SOLUTION: We factor the denominator and then set
where the constants A, B, and C are to be determined. It follows that
x^2 − x + 4 = A(x − 1)(x − 2) + Bx(x − 2) + Cx(x − 1). (2)