11-16 Calculus – Differentiation and Integration
=
∫
+
∫
=
∫
f x dx f x dx
f x dx
a
a
a
a
( ) ( )
( ) .
/
/0
2
2
0
14.
f x dx
f x dx f a x f x
f a x f x
a
a
( )
( ) ( ) ( )
( ) ( ).
/
0
0
2
2
0
∫
=
− =
∫
− = −
if
if
Proof: We have
f x dx f x dx f x dx
f x dx f a x dx
a a
a
a
a
a
a
( ) ( ) ( )
( ) ( )
/
/
/
/
0 0
2
2
0
2
2
∫
=
∫
+
∫
=
∫
+ −
∫∫
− =
=
∫
−
∫
, assuming that
assum
f a x f x
f x dx f t dt
a
a
( ) ( )
( ) ( ) ,
/
/0
2
2
0
iing that and changing the limitsa x t
f x dx f x dx
a
− =
=
∫
+( ) ( )
/
0
2
0
aa a
f x dx
/ /
( ) .
2
0
2
2
∫
=
∫
Similarly,
f x dx f x dx f x dx
f x dx f a x dx
a a
a
a
a
a
a
( ) ( ) ( )
( ) ( )
/
/
/
/
0 0
2
2
0
2
2
∫
=
∫
+
∫
=
∫
− −
∫∫
− = −
=
∫
+
∫
, ( ) ( )
( ) ( ) ,
/
/
assuming that
assu
f a x f x
f x dx f t dt
a
a0
2
2
0
mming that and changing the limitsa x t
f x dx f x dx
a
− =
=
∫
−( ) ( )
/
0
2
00
2
0
a /
.
∫
=
15.
f x dx
f x dx f
f
a
a
a
( )
( )
−
∫
=
∫