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 Assume that we want to evaluate 
P( )
Q( )
∫xx dx
, where P( )
Q( )
x
x
is proper rational function