(II)
Two indefinite integrals with the same derivative lead to the same family of
curves and so they are equivalent Proof Let f and g be two functions such that
( )
d
f x dx
dx ∫
 =
( )
d
g x dx
dx ∫
or
( )
( )
d
f x dx –
g x dx
dx




∫
∫
 = 0
Hence
( )
( )
f x dx –
g x dx
∫
∫
= C, where C is any real number
  (Why )
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