(III)
[
]
( ) +
( )
( )
+
( )
f x
g x
dx
f x dx
g x dx
=
∫
∫
∫
Proof
By Property (I), we have
[ ( ) +
( )]
d
f x
g x
dx
dx




∫
 = f (x) + g(x) (1)
 On the otherhand, we find that
( )
+
( )
d
f x dx
g x dx
dx




∫
∫
 =
( )
+
( )
d
d
f x dx
g x dx
dx
dx
∫
∫
= f (x) + g(x) (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
=
∫