We give the proofs of these properties one by one Proof of P0 It follows directly by making the substitution x = t Proof of P1 Let F be anti derivative of f Then, by the second fundamental theorem of
calculus, we have 
( )
F ( ) – F ( )
–[F ( )
F ( )]
( )
b
a
a
b
f x dx
b
a
a
b
f x dx
=
=
−
= −
∫
∫
Here, we observe that, if a = b, then 
( )
0
a
∫a f x dx =