Integration of rational functions

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

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

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

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