We know how to anti-differentiate a function of the form for any a and b. This will allow us to find the anti-derivative of a rational function if we can reduce it to a sum of terms of that form and possibly a polynomial instead.
If p(x) has a higher degree than q(x) we can extract a quotient polynomial s(x) by a process akin to long division called synthetic division.
We may then be left with a remainder polynomial r(x). We know how to anti-differentiate s(x) so the task of anti-differentiating reduces to anti-differentiating where the numerator has lower degree than the denominator.
Suppose now that we can factor q into factors like (x-a) or (x-b)3 or ((x-d)2+ c2)m.
The wonderful fact is that the expression can be separated into terms each of which has the form or for some a's, d's and c's and integer values of b, each of which can be anti-differentiated.
And here is a procedure for separating it.
Suppose the denominator q can be factored into (x-b)kt(x) such that t(b) is not 0.
And suppose we find the first k terms of the Taylor series expansion of about x = b
Then the terms in that involve inverse powers of (x-b) are given as follows:
If k = 1 there is only one term,
For k = 2 we have
where A is as before while B is
and so on.
There are similar rules for quadratic factors.
The process of separating the denominator q in this manner is called "the method of partial fractions".
We review the various methods again in Section 27.1 and some integrals to practice on are given in Section 27.3. We apologize for the redundancy.