1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integration of rational functions

  1. Mar 6, 2004 #1
    How do I solve the integration of a rational function such as:

    x^2 - 6x - 2
    (x^2 + 2)^2

    If possible, please list the general rule of solving, I DO NOT want the answer, I simply want to know the way of solving it.
    Thanks in advance!
  2. jcsd
  3. Mar 7, 2004 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    The "rule" that you want is "partial fractions".

    If you have a rational function, in which the denominator can be factored into distinct linear factors, such as
    (x- 3)/((x+1)(x-2)), then you can write it as a sum of fractions, each having one factor as denominator:
    (x-3)/((x+1)(x-2))= A/(x+1)+ B/(x-2).
    (Of course, the numerator is of lower degree than the denominator: if not, divide first.)

    If some of the linear factors are repeated, such as
    (x+ 4)/((x+1)2(x-2)), then you will need all powers of that repeated factor: A/(x+1)+ B/(x+1)2+ C/(x-2)

    If some of the factors are quadratics that cannot be factored, then they can, by completing the square, be written in the form "a(x-b)2+ c" and you will need a fraction of the form (Ax+ B)/(a(x-b)2+c), for example (3x2- 2x+ 4)/((x2+ 4)(x+3)) can be written (Ax+B)/(x2+4)+ C/(x+3).

    In this particular example,
    [tex]\frac{x^2-6x- 2}{(x^2+2)^2)^2}[/tex]
    can be written in the form

    Those have to be equal for all x so one way of finding A, B, C, D is by setting those equal:
    [tex]\frac{x^2-6x- 2}{(x^2+2)^2)^2}= \frac{Ax+B}{(x^2+2)^2}+\frac{Cx+D}{x^2+2}[/tex]
    Now multiply both sides by that denominator to clear the fractions and set x equal to 4 different numbers to get 4 equations for A, B, C, and D. You can often choose those numbers to simplify the equations.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?