Infinitesimal calculus MOC

Partial fraction decomposition

Partial fraction decomposition can be used to simplify a complex algebraic fraction, especially for the sake of integration, into a number of simpler partial fractions that are more easily dealt with.

Cases

Denominator with distinct linear factors

𝑓(𝑥)=𝑃(𝑥)(𝑥−𝑎1)⋯(𝑥−𝑎𝑘)=𝐴1𝑥−𝑎1+⋯+𝐴𝑘𝑥−𝑎𝑘

Repeated linear factors

𝑓(𝑥)=𝑃(𝑥)(𝑥−𝑎)𝑐=𝐴1𝑥−𝑎+𝐴1(𝑥−𝑎)2+⋯+𝐴1(𝑥−𝑎)𝑐−1+𝐴1(𝑥−𝑎)𝑐

Irreducible factor

𝑓(𝑥)=𝑃(𝑥)(𝑥−𝑎)(𝑥2+𝑏𝑥+𝑐)=𝐴𝑥−𝑎+𝐵𝑥+𝐶𝑥2+𝑏𝑥+𝑐

Note that the numerator is always one degree lower than the denominator.

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