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 Skip Navigation LinksMath Help > Calculus > Integral > Partial Fractions

Partial Fractions

If you want to integrate f(x)/g(x), where f and g are polynomials, then it is helpful to reduce the degree of the polynomials as much as possible before beginning the integration.

To begin, if the degree of f is greater than or equal to that of g, use Polynomial Division to express f(x)/g(x) as q(x) + r(x)/g(x).  From now on, we'll assume f has smaller degree than g.

If g(x) can be factorized as the product g1(x)e1 g2(x)e2 ... gn(x)en, then f(x)/g(x) can be rewritten as

A1/g1(x) + B1/g1(x)2 + ... + Z1/g1(x)e1  + 
A2/g2(x) + B2/g2(x)2 + ... + Z2/g2(x)e2  + 
   . . .
An/gn(x) + Bn/gn(x)2 + ... + Zn/gn(x)en  + 

Where Ai, Bi, etc. are polynomials of degree smaller than that of gi(x).

The method of finding polynomials Ai, Bi, etc. is to clear the denominators (by multiplying through by the common denominator) then expand all the products, and equate the like terms.  This will give you a set of linear equations in the coefficients of Ai, Bi, etc. which then can be solved using linear algebra.

 . . . . . . webmaster to do:  describe some simple examples of finding Partial Fractions, and add this into the navigation.

Internet references

Wikipedia: Partial Fraction 

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