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Calculus I / Techniques of Integration

Integration by Partial Fractions

A rational function R(x)=P(x)/Q(x)R(x) = P(x)/Q(x) — a ratio of polynomials — can always be integrated by partial-fraction decomposition: breaking RR into a sum of simpler fractions, each of the form A/(ax+b)kA/(ax+b)^k or (Ax+B)/(ax2+bx+c)k(Ax+B)/(ax^2+bx+c)^k, whose antiderivatives are logarithms or arctangents. In finance, bond pricing formulas with rational structure (coupon-bond PV, continuous-payment annuity terms, rational approximations to option prices) regularly produce rational integrands; partial fractions is the systematic machinery that converts them to closed-form expressions. The method decomposes into three mechanical steps: long-divide if the degree of the numerator is not strictly less than the degree of the denominator, factor the denominator, then cover-up or solve for the constants.
Learning objectives. By the end of this note you should be able to:
  • perform polynomial long division on an improper rational function;
  • factor a quadratic or cubic denominator into linear and irreducible-quadratic pieces;
  • write the partial-fraction form for each of the four standard denominator cases;
  • determine the constants by the cover-up method or by comparing coefficients;
  • integrate each piece and assemble the answer.

In this lesson

  1. Preliminary step: long division
  2. The four denominator cases
  3. Finance application: bond pricing with continuous coupons

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