The improper rational functions can be reduced to the proper rational
INTEGRALS         317
functions by long division process Thus, if P( )
Q( )
x
x  is improper, then 
1P ( )
P( )
T( )
Q( )
Q( )
x
x
x
x
x
=
+
,
where T(x) is a polynomial in x and 
1P ( )
Q( )
x
x is a proper rational function As we know
how to integrate polynomials, the integration of any rational function is reduced to the
integration of a proper rational function The rational functions which we shall consider
here for integration purposes will be those whose denominators can be factorised into
linear and quadratic factors