These will be useful in
evaluating the definite integrals more easily P0 :
( )
( )
b
b
a
a
f x dx
f t dt
=
∫
∫
P1 :
( )
–
( )
b
a
a
b
f x dx
f x dx
=
∫
∫ In particular, 
( )
0
a
a f x dx =
∫
P2 :
( )
( )
( )
b
c
b
a
a
c
f x dx
f x dx
f x dx
=
+
∫
∫
∫
P3 :
( )
(
)
b
b
a
a
f x dx
f a
b
x dx
=
+
−
∫
∫
P4 :
0
0
( )
(
)
a
a
f x dx
f a
x dx
=
−
∫
∫
(Note that P4 is a particular case of P3)
P5 :
2
0
0
0
( )
( )
(2
)
a
a
a
f x dx
f x dx
f
a
x dx
=
+
−
∫
∫
∫
P6 :
2
0
0
( )
2
( )
, if
(2
)
( )
a
a
f x dx
f x dx
f
a
x
f x
=
−
=
∫
∫
  and
                 0 if f (2a – x) = – f (x)
P7 :
(i)  
0
( )
2
( )
a
a
a f x dx
f x dx
−
=
∫
∫
, if f is an even function, i e