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 ... g_n(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  + 
   . . .
A_n/g_n(x) + B_n/g_n(x)² + ... + Z_n/g_n(x)^en  + 

Where A_i, B_i, etc. are polynomials of degree smaller than that of g_i(x).

The method of finding polynomials A_i, B_i, 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 A_i, B_i, 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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