Hence, the residues associated with each pole, given by are respectively, and Example 5 (limit method) Limits can be used to find a partial fraction decomposition. Consider the following example: First, factor the denominator which determines the decomposition: Multiplying everything by , and taking the limit when , we get On the other hand, and thus: Multiplying by and taking the limit when , we have and This implies and so . For , we get and thus . Putting everything together, we get the decomposition Example 6 (integral) Suppose we have the indefinite integral: Before performing decomposition, it is obvious we must perform polynomial long division and factor the denominator. Doing this would result in: Upon this, we may now perform partial fraction decomposition. so: . Upon substituting our values, in this case, where x=1 to solve for B and x=-2 to solve for A, we will result in: Plugging all of this back into our integral allows us to find the answer: The role of the Taylor polynomial The partial fraction decomposition of a rational function can be related to Taylor's theorem as follows. Let be real or complex polynomials assume that satisfies Also define Then we have if, and only if, each polynomial is the Taylor polynomial of of order at the point : Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients. Sketch of the proof The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion so is the Taylor polynomial of , because of the unicity of the polynomial expansion of order , and by assumption . Conversely, if the are the Taylor polynomials, the above expansions at each hold, therefore we also have which implies that the polynomial is divisible by For is also divisible by , so is divisible by . Since we then have and we find the partial fraction decomposition dividing by . Fractions of integers The idea of partial fractions can be generalized to other integral domains, say the ring of integers where prime numbers take the role of irreducible denominators. For example: Notes References External links Make partial fraction decompositions with Scilab. Algebra Elementary algebra Partial fractions