This section deals with the integration of general algebraic rational functions, of the form , where and are both polynomials. We already have seen some examples of this form. For example, we know how to integrate functions of the form or or where is a linear factor, is a quadratic factor and is a polynomial of degree . We intend to generalise that previous discussion in this section.
We are assuming the scenario where (the denominator) is decomposible into linear or quadratic factors. These are the only cases relevant to us right now. Any linear or quadratic factor in might also occur repeatedly.
Thus, could be of the following general forms.
( linear factors)
( linear factors; the factor is repeated times)
( linear factors, the factor is repated times)
( linear factors and quadratic factors)
(a particular quadratic factors repeat more than once)
A combination of any of the above.
Suppose that the degree of is and that of is . If we can always divide by to obtain a quotient and a remainder whose degree would be less than .
If is termed a proper rational function.
The partial dfraction expansion technique says that a proper rational function can be expressed as a sum of simpler rational functions each possessing one of the factors of . The simpler rational functions are called partial dfractions.
From now one, we consider only proper rational functions. If is not proper, we make it proper by the procedure described in () above.
Let us consider a few examples.
Let be a product of non-repeated, linear factors:
Then, we can expand in terms of partial dfractions as
where the are all constants that need to be determined
Suppose and . Let us write down the partial dfraction expansion of :
We need to determine , and . Cross multiplying in the expression above, we obtain:
, , can now be determined by comparing coefficients on both sides. More simply since this relation that we’ve obtained should held true for all , we substitute those values of that would straight way give us the required values of , and . These values are obviously the roots of .
Thus, and .
We can therefore write as a sum of partial fractions.
Integrating is now a simple matter of integrating the partial dfractions. This was our sole motive in writing such an expansion, so that integration could be carried out easily. In the example above:
Now, suppose that contains all linear factors, but a particular factor, say , is repeated times.
can now be expanded into partial dfractions as follows:
Comment: partial dfractions corresponding to
This means that we will have terms corresponding to . The rest of the linear factors will have single corresponding terms in the expansion. Here are some examples.