If the degree of P(x)
is less than the degree of Q(x), then the rational function is called proper, otherwise, it
is called improper 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