
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 g_{1}(x)^{e}1 g_{2}(x)^{e}2 ... g_{n}(x)^{e}n, then f(x)/g(x) can be rewritten as
A_{1}/g_{1}(x) + B_{1}/g_{1}(x)^{2} + ... + Z_{1}/g_{1}(x)^{e}1 +
A_{2}/g_{2}(x) + B_{2}/g_{2}(x)^{2} + ... + Z_{2}/g_{2}(x)^{e}2 +
. . .
A_{n}/g_{n}(x) + B_{n}/g_{n}(x)^{2} + ... + Z_{n}/g_{n}(x)^{e}n +
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.
Wikipedia: Partial Fraction
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