Integration of rational functions

Integration of rational functions - Integral of a rational...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Integral of a rational function 1 Intro Most calculus textbooks present this topic as the method of partial fractions . They usually contain a bunch of apparently unrelated examples (sometimes not exhaustive) and do not focus on the purpose of this method. For this reason I have decided to write a few notes about “partial fractions” and I want to focus on what the method is good for and not on examples. Everything that follows is devoted to a “proof” (with numerous and fundamental omissions) of the following statement: Proposition 1.1. Every rational function admits a primitive that can be expressed using elemen- tary functions. This result is proved by presenting a very explicit algorithm for the construction of such a primitive. Finding the primitive of a generic function is a mystery, but the primitive of a rational function is the output of an algorithm!!! Having an algorithm means that we have some kind of machine; we can put a function in it, turn the crank and the output is the result. This specific algorithm has a few drawbacks: first of all it’s very time consuming, integrals that can be solved in a few steps by some other method may require pages of calculations, finally it relies on the knowledge of a factorization of the denominator, but such a factorization might not be so easy to find. In synthesis: If you can reduce an integral to the integral of a rational function you have solved the integral (just turn the crank). If you can choose between this method and another one, choose the other one. 2 Factorization of a polynomial over the real numbers Definition 2.1. A polynomial P ( x ) is a function of this kind: P ( x ) = n X i =0 a i x i If we assume that a n is not zero than n is called the degree of the polynomial. Definition 2.2. A function f ( x ) is called rational if it can be written as a quotient of two poly- nomials: f ( x ) = P ( x ) Q ( x ) The problem that we would like to consider is the following: is it possible to write a polynomial P ( x ) as the product of polynomials of lower degree? Ideally we would like to write P ( x ) as the 1 product of polynomials of the lowest degree possible 1 ; if the degree of P ( x ) is n , we would like to express P ( x ) as the product of n polynomials of degree 1: P ( x ) = n Y i =0 ( c i x + d i ) Unfortunately there are examples of polynomials that cannot be written in this way: x 2 + 1 = x 2- (- 1) = x 2- ( √- 1) 2 = ( x + √- 1)( x- √- 1) We can write this polynomial as the product of two linear factors only if we have a number i such that i 2 =- 1, but no real number satisfies this condition. In general every degree two polynomial ax 2 + bx + c whose discriminant b 2- 4 ac is negative cannot be written as the product of two linear factors. Even if we cannot write a polynomial as the product of linear factors we can still sayfactors....
View Full Document

This note was uploaded on 11/03/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.

Page1 / 6

Integration of rational functions - Integral of a rational...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online