(2)
  Thus, in view of Property (II), it follows by (1) and (2)  that
(
)
( )
( )
f x
g x
dx
+
∫
=
( )
( )
f x dx
g x dx
+
∫
∫ (IV)  For any real number k, 
( )
( )
k f x dx
k
f x dx
=
∫
∫
Proof By the Property (I), 
( )
( )
d
k f x dx
k f x
dx
=
∫ Also
( )
d
k
f x dx
dx




∫
 =  
( )
=
( )
kd
f x dx
k f x
dx ∫
 Therefore, using the Property (II), we have 
( )
( )
k f x dx
k
f x dx
=
∫
∫ (V)
Properties (III) and (IV) can be generalised to a finite number of functions
f1, f2,