Partial Fractions

Real Analysis · Axiom Academy

Break one complicated rational function into a sum of simple pieces — each of which integrates to a clean logarithm or arctangent. 1. One Fraction, Split Into Pieces Start with the textbook case . On its own it is awkward to integrate, but it is secretly the sum of two simple fractions. Finding those two fractions — the partial fractions — turns a hard integral into two easy ones. The single fraction splits into two simple pieces …and each piece integrates straight to a log For each distinct linear factor (x-r) in the denominator, the decomposition gets one term with a single unknown constant on top. To pin down each constant fast, use the cover-up method : cover that factor and evaluate the rest at its root. When a linear factor is repeated , one term is not enough — a factor (x-r)^n contributes a whole tower of terms, one for every power from 1 up to n . Miss a rung and the system has no solution. Coefficient of x : A=3 . Constant: . 4. Irreducible Quadratic Factors A quadratic like x^2+1 that has no real roots cannot be broken any further. Such an irreducible quadratic factor earns a term with a linear numerator — two unknowns, because a quadratic denominator carries two degrees of freedom. (x^2+bx+c) gets : two unknowns. Expanding and matching powers of x : You can now read a denominator's factorization and write down the exact shape of its partial fraction decomposition — then integrate each piece on sight. Scroll up to revisit any step.

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