We have not yet discussed how to integrate rational functions (recall that a rational function is a function of the form f (x)/g(x) , where f , g are polynomials). The method that allows us to do so, in certain cases, is called partial fraction decomposition.
Here we demonstrate this procedure in the case where the denominator g(x) is a product of two distinct linear factors. This method can easily be generalized to the case where g is a product of arbitrarily many distinct linear factors. The cases where g has repeated linear factors or factors of degree 2 are slightly more complicated and will not be considered.
The first step is to divide the polynomial f by the polynomial g to obtain
= h(x) + |
where h(x) and r(x) are polynomials, with the degree of r strictly less than the degree of g . There is a result called the division algorithm that guarantees that we can do this. Since we know how to integrate polynomials, we are left with figuring out how to integrate r(x)/g(x) . Multiplying the numerator and denominator by a constant, we may assume that g(x) is of the form g(x) = (x - a)(x - b) . Since the degree of r is less that 2 , we may write it as r(x) = cx + d .
We want to write r(x)/g(x) in the form
+ |
since we know how to integrate functions of this form (by change of variables, for example). Multiplying the equation
= + |
by (x - a)(x - b) on each side and regrouping terms, we obtain
cx + d | = | A(x - b) + B(x - a) | |
= | (A + B)x + (- Ab - Ba) |
Setting the coefficients of the two polynomials equal to each other, we get a system of two linear equations in the two variables A and B :
A + B | = | c | |
(- b)A + (- a)B = d |
Since a≠b , this system has a solution. Now that we have done all the hard work, we can easily calculate the integral:
dx | = | h(x)dx + dx | |
= | h(x)dx + dx + dx |