Chapter 6: Techniques of Integration
Section 6.4: The Algebra of Partial Fractions
Given the left-hand side of the identity 3x2+1−5x−4=−5⁢x2+3⁢x−17x3−4⁢x2+x−4, the right-hand side is obtained by finding the common denominator x2+1⋅x−4 and adding the two fractions. Given the right-hand side, the left-hand side is called its partial-fraction decomposition. The decomposition is properly applied to a rational function fx=ux/vx for which the degree of u is less than the degree of v. Otherwise, a long division must first be performed so that fx=qx+rx/vx, where q is the quotient, and r is the remainder. The decomposition is then applied to the fraction r/v, not f.
Decomposing a rational function into its partial fractions requires that the denominator of the fraction be factored into nothing worse than quadratics. Indeed, for each nonrepeated linear factor x+A in the denominator of the rational function, its partial fraction decomposition will contain a term of the form a/x+A; and for each nonrepeated quadratic factor x2+B x+C, a term of the form b x+c/x2+B x+C. If a factor σ appears raised to the power n, then the decomposition must contain the sum ∑k=1nλkσk, where λk is ak or bkx+ck, depending on whether σ is linear or quadratic. Table 6.4.1 illustrates the structure of the partial-fraction decomposition in several explicit cases of factored denominators vx.
ax+A+b1x+c1x2+B x+C+b2x+c2x2+B x+C2
Table 6.4.1 Decomposition templates for different denominators of f=u/v
To a linear factor (simpler than the quadratic) there corresponds a partial fraction with a "simple" numerator, namely, a constant; to a quadratic factor (messier than the linear), a "messy" numerator, namely, the linear expression b x+c. This observation is the basis for the author's own mnemonic: "Simple-simple; messy-messy, but watch out for the repeats." Simple (linear) factors require simple numerators, but messy (quadratic) factors require "messy" numerators. The repeated-factor rule is the one giving students most difficulty, and the irony in the mnemonic is the caution against "repeats" in the midst of a sentence that has multiple repeats.
A final issue to consider is the number field over which the denominator is factored. Working over the reals, the factor x2+1 is irreducible, and would be considered a quadratic factor. But working over the complex numbers, every polynomial is the product of linear factors, even if some of the linear factors are repeated. Controls engineers typically work over the complex field, so all factors would be considered linear. In an integral calculus course, a quadratic factor that was irreducible over the reals would typically remain a quadratic factor.
Long Division of Polynomials
In Example 6.3.13 the identity u2u2−4=1+4u2−4=1+1u−2−1u+2 is a key ingredient of an integration following upon a trig substitution. The middle of the identity is a result of a long division; and the end, of a partial-fraction decomposition. Unfortunately, Maple does not support the standard formatting of a long division. Table 6.4.2 contains an approximation to the tableau that might appear in a printed textbook.
u2−4 ) u2
Table 6.4.2 Maple's approximation to the long-division tableau
The u2 in u2−4, the divisor, is divided into the u2 in the dividend. The quotient is 1. The quotient is then multiplied by the divisor, and the product, u2−1, is written in the penultimate line. This product is subtracted from the dividend, the remainder being 4. Hence, by long division, u2u2−4=1+4u2−4.
Execute the long division implied by the fraction x4−7 x2+5 x−8x2−2 x+3.
Obtain the partial-fraction decomposition of 7⁢x−23x2−7⁢x+12.
Obtain the partial-fraction decomposition of 7⁢x3−86⁢x2+346⁢x−455x4−15⁢x3+84⁢x2−208⁢x+192.
Obtain the partial-fraction decomposition of 5⁢x3−11⁢x2+18⁢x+1x4−5⁢x3+14⁢x2−19⁢x+15.
Obtain the partial-fraction decomposition of 5⁢x4+37⁢x3+15⁢x2−150⁢x+109x5+11⁢x4+21⁢x3−59⁢x2−21⁢x+49.
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