1
INTEGRALS         293
Proof Let F be any anti derivative of f, i e ,
dF( )
x
dx
 = f (x)
Then
( )
∫f x dx
 = F(x) + C
Therefore
( )
d
dx ∫f x dx
 =
(
F ( ) + C)
d
x
dx
=
F ( ) =
( )
d
x
f x
dx
Similarly, we note that
f ′(x) =
( )
d f x
dx
and hence
f( )
∫′x dx
 = f (x) + C
where C is arbitrary constant called constant of integration (II)
Two indefinite integrals with the same derivative lead to the same family of
curves and so they are equivalent