)
or
( )
∫f x dx
 = 
( )
g x dx +C
∫
So the families of curves {
}
1
1
( )
C ,C
R
f x dx +
∈
∫
and
{
}
2
2
( )
C , C
R
g x dx +
∈
∫
 are identical Hence, in this sense, 
( )
and
( )
f x dx
g x dx
∫
∫
 are equivalent 294
MATHEMATICS
� Note The equivalence of the families {
}
1
1
( )
f x dx+ C ,C
∈
∫
R  and
{
}
2
2
( )
g x dx+ C ,C
∈
∫
R  is customarily expressed by writing 
( )
=
( )
f x dx
g x dx
∫
∫
,
without mentioning the parameter (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)