Integrating rational functions

In summary, the conversation discusses the concept of partial fraction integration for rational functions and how it can be used to solve integrals. It also touches on the issue of irreducible quadratic factors in the denominator and how they can be handled using methods like u-substitution and trigonometric substitution. The conversation ends with a solution for integrating the form c/qm in the general case.
  • #1
suffian
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 occurring 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?
 
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  • #2
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!
 
  • #3
Complete the square, trig substitution.
 
  • #4
yeah, that seems to work well enough. it was actually just the next section, but thanks a lot.
 

What are rational functions?

Rational functions are functions that can be expressed as a ratio of two polynomials, where the denominator is not equal to zero. They are also known as ratio functions.

What is the process for integrating rational functions?

The process for integrating rational functions involves finding the antiderivative of the function. This can be done using techniques such as partial fractions, substitution, or integration by parts.

What is a partial fraction decomposition?

Partial fraction decomposition is a method used to write a rational function as a sum of simpler fractions. This allows for easier integration of the rational function.

Can all rational functions be integrated?

No, not all rational functions can be integrated. If the function has a denominator that cannot be factored or if the degree of the numerator is greater than or equal to the degree of the denominator, then the function cannot be integrated using elementary methods.

Why is integrating rational functions important?

Integrating rational functions is important because it allows us to solve a wide range of problems in fields such as physics, engineering, and economics. It also helps us to understand the behavior of functions and their relationship with other mathematical concepts.

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