1. Properties Of
Definite Integral

2. Properties Of
Definite Integral
1 a
b
n 1
b
x 
x dx  
n  1 a


n

3. Properties Of
Definite Integral
1 a
b
n 1
b
x 
x dx  
n  1 a


n
2 a kf  x dx k a f  x dx
b
b
can only factorise constants 

4. Properties Of
Definite Integral
1 a
b
n 1
b
x 
x dx  
n  1 a


n
2 a kf  x dx k a f  x dx
b
b
can only factorise constants 
3 a  f  x   g  x dx  a f  x dx  a g  x dx
b
b
b

5. Properties Of
Definite Integral
1 a
b
n 1
b
x 
x dx  
n  1 a


n
can only factorise constants 
2 a kf  x dx k a f  x dx
b
b
3 a  f  x   g  x dx  a f  x dx  a g  x dx
b
b
b
4 a f  x dx  a f  x dx  c f  x dx
b
c
b

6. 5 a x n dx  0
b

7. 5 a x n dx  0
b
, if f  x   0 for a  x  b

8. 5 a x n dx  0
b
0
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b

9. 5 a x n dx  0
b
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b
0
6 a f  x dx  a g  x dx
b
b

10. 5 a x n dx  0
b
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b
0
6 a f  x dx  a g  x dx
b
b
, if f  x   g  x  for a  x  b

11. 5 a x n dx  0
b
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b
0
6 a f  x dx  a g  x dx
b
b
7 a f  x dx   b f  x dx
b
a
, if f  x   g  x  for a  x  b

12. 5 a x n dx  0
b
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b
0
6 a f  x dx  a g  x dx
b
b
, if f  x   g  x  for a  x  b
7 a f  x dx   b f  x dx
b
a
8  f  x   0 ,
a
a
if f  x  is odd

13. 5 a x n dx  0
b
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b
0
6 a f  x dx  a g  x dx
b
b
, if f  x   g  x  for a  x  b
7 a f  x dx   b f  x dx
b
a
a
8  f  x   0 ,
a
a
a
a
0
if f  x  is odd
9  f  x   2 f  x  ,
if f  x  is even

14. 5 a x n dx  0
b
, if f  x   0 for a  x  b
, if f  x   0 for a  x  b
0
6 a f  x dx  a g  x dx
b
b
, if f  x   g  x  for a  x  b
7 a f  x dx   b f  x dx
b
a
a
8  f  x   0 ,
a
a
a
a
0
if f  x  is odd
9  f  x   2 f  x  ,
if f  x  is even
NOTE :
odd  odd  even
odd  even  odd
even  even  even

15. 2
e.g. (i)  6 x 2 dx
1

16. 2
e.g. (i)  6 x 2 dx
1
2
1 x 3 
 6
 3 1


17. 2
e.g. (i)  6 x 2 dx
1
2
1 x 3 
 6
 3 1

 223  13 
 14

18. 2
e.g. (i)  6 x 2 dx
1
5
ii   3 xdx
0
2
1 x 3 
 6
 3 1

 223  13 
 14

19. 2
e.g. (i)  6 x 2 dx
1
2
1 x 3 
 6
 3 1

 223  13 
 14
5
5
0
0
1
3
ii   3 xdx   x dx

20. 2
e.g. (i)  6 x 2 dx
1
2
1 x 3 
 6
 3 1

 223  13 
 14
5
5
0
0
1
3
ii   3 xdx   x dx
5
3 
 x 
4  0
4
3

21. 2
e.g. (i)  6 x 2 dx
1
2
1 x 3 
 6
 3 1

 223  13 
 14
5
5
0
0
1
3
ii   3 xdx   x dx
5
3 
 x 
4  0
3 3 5
 x x 0
4
4
3

22. 2
e.g. (i)  6 x 2 dx
1
2
1 x 3 
 6
 3 1

 223  13 
 14
5
5
0
0
1
3
ii   3 xdx   x dx
5
3 
 x 
4  0
3 3 5
 x x 0
4
3 3
 5 5  0 
4
153 5

4
4
3

23. 2
iii   sin 5 xdx
2

24. 2
iii   sin 5 xdx  0
2
odd function 5  odd function

25. 2
iii   sin 5 xdx  0
2
1
iv  x 3  2 x 2  x  1dx
1
odd function 5  odd function

26. 2
iii   sin 5 xdx  0
odd function 5  odd function
2
1
1
1
0
iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx

27. 2
iii   sin 5 xdx  0
odd function 5  odd function
2
1
1
1
0
iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx
1
 2 x3  x
 2
0
3


28. 2
iii   sin 5 xdx  0
odd function 5  odd function
2
1
1
1
0
iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx
1
 2 x3  x
 2
0
3

 2 1 3  1  0
 2  

3

10

3

29. 2
iii   sin 5 xdx  0
odd function 5  odd function
2
1
1
1
0
iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx
1
 2 x3  x
 2
0
3

 2 1 3  1  0
 2  

3

10

3
Exercise 11C; 1bce, 2adf, 3ab (i, iii), 4bcf, 5, 6ac, 7df, 8b,
12b, 13*