# Integrating rational functions

suffian

## Main Question or Discussion Point

I've just finished reading the section on partial fraction integration from my text. The book describes how all rational functions can be integrated by performing a partial fraction decomposition and subsequently integrating the partial fractions using methods that are already known. I tried to verify this fact for myself, but I ran into some trouble with irreducible quadratic factors occuring in the denominator of the rational function (the linear factors look like they could all just be "ln-ified").

As shown in the text, an irreducible quadratic factor q = dx2 + ex + f raised to n in the denominator of a rational function needs to be taken apart as follows:
(a1x+b1)/q + (a2x+b2)/q2 + ... + (an-1x+bn-1)/qn-1 + (anx+bn)/qn

So the problem of integrating all rational functions relegates to solving integrals of the following form:
(ax+b)/qm (m is natural number)

With a little insight, you can reduce this a little further by adding and subtracing a constant to make it partially amenable to a u-substitution:
(ax+b +c -c)/qm (c such that ax+b+c is k*q')
(ax+b+c)/qm - c/qm (where left integral can be solved with u-sub)

I don't see how to reduce this last form (c/qm where c constant and q quadratic) despite the claim by the book that it is integrable using methods already known. Does anyone else know (or can figure out) how it's possible to integrate this form in the general case?

Hurkyl
Staff Emeritus
Gold Member
Although m must be an integer in your problem, consider how you would integrate 1/q^(1/2).

The next post is another hint, but try going by just this one if you can!

Hurkyl
Staff Emeritus
Gold Member
Complete the square, trig substitution.

suffian
yeah, that seems to work well enough. it was actually just the next section, but thanks a lot.