Editing Partial fraction decomposition (section)

==Application to symbolic integration==

For the purpose of [[symbolic integration]], the preceding result may be refined into

{{math_theorem|name=Theorem|Let ''f'' and ''g'' be nonzero polynomials over a field ''K''. Write ''g'' as a product of powers of pairwise coprime polynomials which have no multiple root in an algebraically closed field:

<math display="block">g=\prod_{i=1}^k p_i^{n_i}.</math>

There are (unique) polynomials ''b'' and ''c''<sub>''ij''</sub> with {{math|deg ''c''<sub>''ij''</sub> < deg ''p''<sub>''i''</sub>}} such that
<math display="block">\frac{f}{g} = b+\sum_{i=1}^k\sum_{j=2}^{n_i}\left(\frac{c_{ij}}{p_i^{j-1}}\right)' + \sum_{i=1}^k \frac{c_{i1}}{p_i}.</math>
where <math> X'</math> denotes the derivative of <math>X.</math>}}

This reduces the computation of the [[antiderivative]] of a rational function to the integration of the last sum, which is called the ''logarithmic part'', because its antiderivative is a linear combination of logarithms. 

There are various methods to compute decomposition in the Theorem. One simple way is called [[Charles Hermite|Hermite]]'s method. First, ''b'' is immediately computed by Euclidean division of ''f'' by ''g'', reducing to the case where deg(''f'') < deg(''g''). Next, one knows deg(''c''<sub>''ij''</sub>) < deg(''p''<sub>''i''</sub>), so one may write each ''c<sub>ij</sub>'' as a polynomial with unknown coefficients. Reducing the sum of fractions in the Theorem to a common denominator, and equating the coefficients of each power of ''x'' in the two numerators, one gets a [[system of linear equations]] which can be solved to obtain the desired (unique) values for the unknown coefficients.