1. Odd & Even Functions
(1) Even f  x   f  x 

2. Odd & Even Functions
(1) Even f  x   f  x 
a a
 f  x dx  2 f  x dx
a 0

3. Odd & Even Functions
(1) Even f  x   f  x 
a a
 f  x dx  2 f  x dx
a 0
ca ca
NOTE: horizontal shift
 f  x  c dx  2  f  x  c dx
ca c

4. Odd & Even Functions
(1) Even f  x   f  x 
a a
 f  x dx  2 f  x dx
a 0
ca ca
NOTE: horizontal shift
 f  x  c dx  2  f  x  c dx
ca c
(2) Odd f  x    f  x 

5. Odd & Even Functions
(1) Even f  x   f  x 
a a
 f  x dx  2 f  x dx
a 0
ca ca
NOTE: horizontal shift
 f  x  c dx  2  f  x  c dx
ca c
(2) Odd f  x    f  x 
a
 f  x dx  0
a

6. Odd & Even Functions
(1) Even f  x   f  x 
a a
 f  x dx  2 f  x dx
a 0
ca ca
NOTE: horizontal shift
 f  x  c dx  2  f  x  c dx
ca c
(2) Odd f  x    f  x 
a
 f  x dx  0
a
ca
NOTE: horizontal shift
 f  x  c dx  0
ca

7. Odd & Even Functions
(1) Even f  x   f  x 
a a
 f  x dx  2 f  x dx
a 0
ca ca
NOTE: horizontal shift
 f  x  c dx  2  f  x  c dx
ca c
(2) Odd f  x    f  x 
a
 f  x dx  0
a
ca
NOTE: horizontal shift
 f  x  c dx  0
ca
a a
 f  x dx   f a  x dx
(3)
0 0

8. a
 f a  x dx
Proof:
0

9. a
 f a  x dx
Proof: u ax
0 du  dx

10. a
 f a  x dx
Proof: u ax
0 du  dx
x  0, u  a
x  a, u  0

11. a
 f a  x dx
Proof: u ax
0 0 du  dx
   f u du x  0, u  a
a
x  a, u  0

12. a
 f a  x dx
Proof: u ax
0 0 du  dx
   f u du x  0, u  a
a
a x  a, u  0
  f u du
0

13. a
 f a  x dx
Proof: u ax
0 0 du  dx
   f u du x  0, u  a
a
a x  a, u  0
  f u du
0
a
  f  x dx
0

14. a
 f a  x dx
Proof: u ax
0 0 du  dx
   f u du x  0, u  a
a
a x  a, u  0
  f u du
0
a
  f  x dx
0
odd  odd  even
odd  even  odd
even  even  even

15. 1
e.g. i   sin 3 xdx
1

16. 1
e.g. i   sin 3 xdx odd function 3  odd function
1

17. 1
e.g. i   sin 3 xdx  0 odd function 3  odd function
1

18. 1
e.g. i   sin 3 xdx  0 odd function 3  odd function
1
1
ii  x 2 1  xdx
0

19. 1
e.g. i   sin 3 xdx  0 odd function 3  odd function
1
1 1
ii  x 2 1  xdx   1  x 2 xdx
0 0

20. 1
e.g. i   sin 3 xdx  0 odd function 3  odd function
1
1 1
ii  x 2 1  xdx   1  x 2 xdx
0 0
1
 1 3 5

   x  2 x  x dx

2 2 2
0 

21. 1
e.g. i   sin 3 xdx  0 odd function 3  odd function
1
1 1
ii  x 2 1  xdx   1  x 2 xdx
0 0
1
 1 3 5

   x  2 x  x dx

2 2 2
0 
7 1
2 3
4 2  5
 x  x  x 
2 2 2
3 5 7 0

22. 1
e.g. i   sin 3 xdx  0 odd function 3  odd function
1
1 1
ii  x 2 1  xdx   1  x 2 xdx
0 0
1
 1 3 5

   x  2 x  x dx

2 2 2
0 
7 1
2 3
4 2  5
 x  x  x 
2 2 2
3 5 7 0
2 4 2
   0
3 5 7
16

105

23. Exercise 2I; 1 bdf, 2 ace, 3
Exercise 2J; 42, 44
The 100 (not 78)