11 x1 t16 02 definite integral (2013)

Properties Of
Definite Integral

Properties Of
Definite Integral
1 a

b

n 1

b

x 
x dx  
n  1 a


n

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 

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


3 a  f  x   g  x dx  a f  x dx  a g  x dx
b

b

b

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

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

5 a x n dx  0
b

, if f  x   0 for a  x  b

5 a x n dx  0
b

0

, if f  x   0 for a  x  b
, if f  x   0 for a  x  b

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

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

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

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

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


9  f  x   2 f  x  ,

if f  x  is even

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


9  f  x   2 f  x  ,

if f  x  is even

NOTE :
odd  odd  even
odd  even  odd
even  even  even

2

e.g. (i)  6 x 2 dx
1

2

1 x 3 
 6
 3 1


2

e.g. (i)  6 x 2 dx
1

2

1 x 3 
 6
 3 1

 223  13 
 14

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

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

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

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

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

2

iii   sin 5 xdx
2

2

iii   sin 5 xdx  0
2

odd function 5  odd function

2

2

1

iv  x 3  2 x 2  x  1dx
1


2



2

1

1

1

0

iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx

2



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



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

2



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*

11 x1 t16 02 definite integral (2013)

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11 x1 t16 02 definite integral (2013)