Integration of Rational Functions

 

 

In a Nut Shell:  Recall that a rational function is simply the ratio of two polynomials,

say  P(x) and Q(x).   Let  R(x) be the following rational function.

                

                                   R(x) = P(x) / Q(x)

 

where P(x) and Q(x) are polynomials

 

 

 

The strategy involved in evaluating integrals of rational functions is then to

express the rational function as a sum of simpler functions called partial fractions.   

Thus the method of integration is called the method of “partial fractions”.

 

 

 

Note:  If P(x) is of equal or higher degree than Q(x) first divide Q(x) into P(x) using

long division so that P(x) is of lower degree than Q(x).

 

The result for the rational function, R(x),  is:

 

   R(x) = p(x) + P(x)/Q(x)  =  p(x) + F1(x) + F2(x) + . . .

 

  The  Fi(x), “the partial fractions”, commonly have two forms that are useful.

 

   a.    A/(ax + b)n

 

   b.   (Bx + C)/(ax2+ bx + c)n

 

       with   b2 – 4ac < 0   in the quadratic term

 

If  the exponent, n, is such that n  > 1, then the decomposition must account for the

multiplicity of these terms.      i.e.  for forms as in part a,

 

If n = 2, then   P(x)/Q(x)  =  A1/(ax + b)  +  A2/ax + b)2

 

A similar situation holds for multiplicity of quadratic forms as in part b.  i.e. If n = 2,

 

P(x)/Q(x) =  (B1(x) + C1)/(ax2+ bx + c) 

                                        + (B2(x) + C2)/(ax2+ bx + c)2

 

 

Click here for examples.

 


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