1. Properties Of
Definite Integral

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

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

4. Properties Of
Definite Integral
n 1 b
x 
1 a
b
x dx  
n
 n  1 a

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
n 1 b
x 
1 a
b
x dx  
n
 n  1 a

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
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
, if f  x   0 for a  x  b
0 , if f  x   0 for a  x  b

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

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

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

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

15. 2
e.g. (i)  6 x 2 dx
1

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

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

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

19. 2 2
e.g. (i)  6 x 2 dx 1 x 3 
 6
1  3 1 
 223  13 
 14
5 5 1
ii   3 xdx   x dx3
0 0

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

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

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

23. 2
iii   sin 5 xdx
2

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

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

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

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

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

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