where P and Q are polynomials. If the degree of P is less than the degree of Q, then f is proper; if not, then we can use long division to write
where S is polynomial, and the degree of R is less than the degree of Q. Since we certainly know how to integrate any polynomial, S poses no problem; so the problem of integrating rational function reduces to the problem of integrating proper rational functions.
To integrate any rational
function, we need to know how to factor any polynomial. The important
fact is that every polynomial can be factored into linear factors - terms like
ax+b - if we allow complex numbers for a. If we stick with real numbers,
then every polynomial can be factored into linear terms and quadratic terms (of
the form 
). We'll stick with reals, so we need to worry about the
quadratic terms.
The quadratic terms arise from pairs of complex roots (called
conjugates). For example, the polynomial 
can't be factored over the
real numbers (we often say that ``it can't be factored''); in fact, however,
the polynomial can be factored, but it has complex roots:
where 
.
Fundamental theorem of algebra: every 
degree polynomial p
with real coefficients can be factored into n linear terms of the form
, where 
is a complex number:
Consider proper rational function f
We consider four cases:
Then
Of course, each of the terms on the right are integrated to yield logs, so we know how to integrate such rational functions.
Then for each repeated term, we need to add a term for each power to the partial fraction:
Since the higher powers (beyond linear) are easily integrated using u-substitution, again, we see that these rational functions can be integrated.
Then the partial fraction decomposition of Q will contain a term of the form
Such terms are integrated using the arctan:
(you may need to complete a square first). Again, this class of rational functions can be integrated.
Then the partial fraction decomposition of Q will contain terms of the form
Such terms are integrated using the arctan:
(you may need to complete a square first). Again, this class of rational functions can be integrated.
In this section the theory of analytical integration of rational functions is presented. The claim, substantiated by example, is that any rational function can be written as a sum of so-called ``partial fractions'' (actually fractional expressions), each of which we know how to integrate.
This is another one of those sections which is ``at risk'' in today's reformed calculus environment: there was a time when these techniques were important for the solution of real problems - that's not the case anymore. Because this is an algorithmic technique, any computer can solve these integrals faster and more accurately than you can! Just enjoy the algebraic notions which are developed in the section....