EXAMPLE 39
EXAMPLE 40
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.
Examples 42–47 are BC ONLY.
EXAMPLE 42