Definite Integrals
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
Calculus II
7
Definite Integrals
2
Properties of Definite Integrals
7
Integration by Parts
15
Continuity and Differentiability of a Definite Integral
18
Improper Integrals
23
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1
F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Definite Integrals
Definition
Let f (x) be a continuous function defined on [a, b] divide the interval by the points
a  x0  x1  x 2    x n 1  b from a to b into n subintervals. (not necessarily equal
width) such that when n   , the length of each subinterval will tend to zero.
In the ith subinterval choose  i  [ x i 1 , xi ] for i  1,2,  , n . If lim ( xi  xi 1 ) f ( i ) exists
n 
and is independent of the particular choice of x i and  i , then we have

b
a
Remark
f ( x) dx  lim
n 
n
 (x  x
i
i 1
i 1
) f (i )
For equal width, i.e. divide [a, b] into n equal subintervals of length, i.e. h 
we have

b
a
f ( x) dx  lim
n 
n

i 1
f ( i )h  lim
n
n
 f ( )
i 1
i
ba
,
n
ba
.
n
Choose  i  xi and x i  a  ih


b
a
OR

b
a
f ( x) dx  lim
n 
f ( x) dx  lim
n 
n
 f (a 
i 1
n 1
 f (a 
i 0
ba
i) h
n
ba
i) h
n
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
1
 2  1  2
 n 1 
1  1    1      1 


n  n
n 

 n  n


2
Example
Evaluate lim
Example
I
Example
Using a definite integral, evaluate
(a)
=

1
1
1
lim 


n  
2
2
2
2
2
n 2
n  n2
 n 1
1
1 
 1
lim 

 
2n 
 n 1 n  2
lim
n
2








(ln 2)
n 
n
(b)
2
1
( )
e
n!
n
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
1
AL95II
 ax  bx  1 x
 , where a, b  0 .
(a) Evaluate lim 
x 0
3


(b) By considering a suitable definite integral, evaluate
 13
23
n3 

lim 
 3
 3
n   n3  13
n  23
n  n3 

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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
AL83II-1
Evaluate
(a)
(b)
(c)
dx


(x  a)( x  b)

,
ln( 1  tan x)dx , [ Hint: Put u 

 x .]
4
1

2
(n  1)  
lim cos  cos
   cos

n  n
n
n
n 

4
0
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
1
Example*
I
=
 (n 3 1)( n 3  2 3 )( n 3  3 3 )  (n 3  n 3 )  n
lim 

n 
n 3n


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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Properties of Definite Integrals
P1
The value of the definite integral of a given function is a real number, depending on its lower
and upper limits only, and is independent of the choice of the variable of integration, i.e.

b
a
P2

b
P3

a
P4
Let a  c  b , then
Example
(a)
a
a
b
b
a
a
f ( x)dx   f ( y)dy   f (t )dt .
a
f ( x)dx   f ( x)dx
b
b
b
b
a
f ( x)dx   f ( x)dx   0 dx  0

3
0
[x]dx

b

3
a
(b)
0
c
b
a
c
f ( x)dx   f ( x)dx   f ( x)dx
x[x]dx
(c)

2
1
x  1dx
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
P5*
Comparison of two integrals
If f ( x)  g ( x) x  (a, b) , then
Example
1
x
0
Prove that
2
a
b
f ( x)dx   g ( x)dx
a
1
dx   x dx .
0

(a)

2
0

(b)
Example
b
x 2  x , for all x  (0,1) ;
hence
Example


2
0
3
2   2
x sin x dx    .
32

sin xdx   2 sin n 1 xdx
n
0
In Figure, SR is tangent to the curve y  ln x at x  r , where r  2 .
By considering the area of PQRS , show that
1
2
1
r
2

Hence show that

n
1
2
r
ln xdx  ln r
ln xdx  ln( n! ) 
1
ln n for any integer n  2 .
2
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
P6
Rules of Integration
If f ( x), g ( x) are continuous function on [a, b] then
P7*
b
b
a
a
(a)
 kf ( x)dx  k 
(b)
  f ( x)  g ( x)dx  
(a)
(b)
(c)
(d)
f ( x)dx for some constant k.
b
b
a
a

a

a

2a

b
a
0
Proof
(a)
Example
Evaluate (a)
a : any real constant.
f ( x) dx    f ( x)  f ( x) dx .
a
a
a
a
f ( x)dx   f (a  x)dx .
0
0
b
f ( x)dx   g ( x)dx .
0
f ( x)dx    f ( x)  f (2a  x) dx
a
0
b
f ( x)dx   f (a  b  x)dx
a
x4
0 x 4  ( x  b)4 dx
b

(b)

3
ln(tan x)dx
6
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Exercise 7C
5.
By proving that
evaluate
6

a
0

0

(a)
(a) Show that
a
f ( x)dx   f (a  x)dx

2
0
a
a
(cos x  sin x)1997 dx
(b)


2
0
sin x
dx
sin x  cos x
f ( x) dx    f ( x)  f ( x) dx
a
0
(b) Using (a), or otherwise, evaluate the following integrals:

(i)
d
  1  sin 
4

(iii)
1

1
ln( x  1  x 2 )dx
4
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Remind
sin x, e x  e x are odd functions.
cos x,
1
are even functions.
x 1
2
Graph of an odd function
P8
(i)
If f ( x)  f ( x) (Even Function)
then
(ii)
Graph of an even function

a
a
a
f ( x) dx  2 f ( x) dx
0
If f ( x)   f ( x) (Odd Function)
then

a
a
f ( x) dx  0
Proof
Example
Evaluate (a)
e x  e x
 sin( 2 ) dx
Example
Prove that (a)
 sin xdx  0



(b)
(b)
1

1
(e x  e x ) sin xdx
a
sin x
a
1  x2

dx  0
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Definition
Let S be a subset of R , and f (x) be a real-valued function defined on S . f (x) is called a
PERIODIC function if and only if there is a positive real number T such that f ( x  T )  f ( x) ,
for x  S . The number T is called the PERIOD.
P9
If f (x) is periodic function, with period T i.e. f ( x  T )  f ( x)

 T
(a)

f ( x)dx  
(b)


f ( x)dx  
(c)

(d)

0
 T
T 
nT
0
 T
T
f ( x)dx
f ( x)dx
T
f ( x)dx   f ( x)dx
0
T
f ( x) dx  n f ( x) dx for n  Z 
0
Proof
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Theorem
Cauchy-Inequality for Integration
If f (x) , g (x ) are continuous function on [a, b] , then
2
2
2
 b f ( x) g ( x)dx    b  f ( x) dx  b g ( x) dx 

 a
  a
 a

Proof
Example


2
0
x sin x dx 

2 2
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Theorem
Triangle Inequality for Integration

b
a
f ( x) dx 

b
a
f ( x) dx
Proof
Example
Show that

1
3
e x sin x

dx 
2
x 1
12e
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Integration by Parts
Theorem
Integration by Parts
Let u and v be two functions in x . If u ' ( x) and v' ( x) are continuous on a, b , then
 uv' dx  uv   vu' dx
b
b
b
a
a
a
 udv  uv   vdu
or
b
b
b
a
a
a
(a)
n
 ln xdx  ln n e
n  n 1
Example
Show that
.
AL84II-1
(a) For any non-negative integer k, let uk  
1
(b)

0
1
 xe dx  1
x
0
sin kx
dx .
sin x
Express uk  2 in terms of uk .
Hence, or otherwise, evaluate uk .

(b) For any non-negative integers m and n , let I (m, n)   2 cos m  sin n dθ
0
(i)
Show that if m  2 , then
I (m, n)
=
 m 1

 I (m  2, n  2) .
 n 1 
(ii)
Evaluate I (1, n) for n  0 .
(iii)
Show that if n  2 , then
I (0, n)
(iv)
=
 n 1

 I (0, n  2) .
 n 
Evaluate I (6,4) .
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics

AL94II-11
For any non-negative integer n , let
n 1
I n   4 tan n x dx
0
(a) (i)
1  
Show that
 
n 1  4 
[ Note:
You may assume without proof that x  tan x 
 In 
1  
 .
n 1  4 
4x


for x  [0, ] .
4
]
(ii) Using (i), or otherwise, evaluate lim I n .
n 
(iii) Show that I n  I n  2 
1
for n  2,3,4, .
n 1
(1)k 1
.
k 1 2k  1
n
(b) For n  1,2,3,  , let an  
(i)
Using (a)(iii), or otherwise, express an in terms of I 2 n .
(ii) Evaluate lim a n .
n
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics


a sin x  b cos x
dx .
sin x  cos x
Example
I
Solution
Let u 
Example
Show that
2
0

2

( ans:
4
( a  b) )
x
(a)
(b)


2
0


ln(sin x)dx   2 ln(cos x)dx 
0

2
0

1 2

ln(sin 2 x)dx  ln 2

2 0
4

ln(sin 2 x)dx 
Deduce that

1 
ln(sin x)dx   2 ln(sin x)dx
0
2 0

2
0
ln(sin x)dx 

2
ln
1
2
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Continuity and Differentiability of a Definite Integral
Theorem
Mean Value Theorem for Integral
If f (x) is continuous on [a,b] then there exists some c in [a,b] and

b
a
f ( x)dx  f (c)(b  a)
Proof
Theorem
Continuity of definite Integral
If f (t ) is continuous on a, b and let A( x)   f (t )dt x  [ a, b] then A(x ) is
x
a
continuous at each point x in a, b .
Proof
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Theorem *
Fundamental Theorem of Calculus
x
Let f (t ) be continuous on [a,b] and F ( x)   f (t )dt . Then F ' ( x)  f ( x) , x  (a, b)
a
Proof
Remark :
H ( x)  
g ( x)
a
f (t )dt  H ' ( x)  f ( g ( x)) g ' ( x)
Proof
Example
Evaluate the Derivatives of the following
x 2 1
(i)
H ( x)  
(ii)
H ( x)   xf (t )dt
(iii)
H ( x)  
0
t 2dt
x
0
x2
x
f (t )dt
Solution
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Example
Let f : R  R be a function which is twice-differentiable and with continuous second
x
t
derivative. Show that f ( x)  f (0)  xf ' (0)     f ' ' ( s)ds dt , x  R .
0 0

x5
(1) n x 4 n 1
.

5
4n  1
x dx
Prove that lim S n ( x)  
.
0 1  x4
n 
Example
Let Sn ( x)  x 
AL90II-5
(a) Evaluate
d xn
f (t ) dt , where f is continuous and n is a positive integer.
dx 0
x2
(b) If F ( x)   3 et dt , find F ' (1).
2
x
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Example
Evaluate

lim
x
0
x 0
cos t 2 dt
x
1
1 x 2
AL97II-5(b) Evaluate lim  3  et dt  2  .
0
x 0  x
x 
AL98II-2
Let f : R  R be a continuous periodic function with period T .
x
d  x T
 0 f (t )dt  0 f (t )dt 

dx 
(a)
Evaluate
(b)
Using (a), or otherwise, show that

x T
x
T
f (t )dt   f (t )dt for all x .
0
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Example
Let k be a positive integer.
Evaluate
(a)
(b)
(c)
Example
d x
cos t 2 dt
dx 0
d y 2k
cos t 2 dt

0
dy
lim
y 0
1
y 2k

y 2k
0
cos t 2 dt
Suppose f (x) has a continuous derivative on [0,1]. If
1

0
f (tx)dt  f ( x)  x 2 for all x and
f (1)  0, find f (x ) .
Remark
1

0
f (tx)dt is a function of x and so
d 1
f (tx)dt  0
dx 0
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Improper Integrals
Definition
A definite integral

b
f ( x)dx is called IMPROPER INTEGRAL if the interval [a,b] of
a
integration is infinite, or if f (x) is not defined or not bounded at one or more points in [a,b].

Example

Definition
(a)


(b)

b
(c)


0


1


1

f ( x)dx is defined as lim

f ( x)dx is defined as lim

e dx ,
a
 
tan xdx ,

x

e
x2
dx ,
0
0
b
b   a
b
a   a
f (x)dx is defined as
( Or lim

c
   
f ( x)dx  lim

c


-
   c
1
dx are improper integral.
x2
f ( x)dx
f ( x)dx
f ( x)dx  

c
f ( x)dx for any real number c .
f ( x)dx )
(d) If f (x) is continuous except at a finite number of points, say a, x1, b where

a  x1  b , then
b
a
f ( x)dx is defined to be

c

d
lim  f ( x)dx  lim   f ( x)dx  lim   f ( x)dx  lim  f ( x)dx
 a

  x1
c
  x1
 b

d
for any c, d such that a  c  x1  d  b .
Definition
The improper integral is said to be Convergent or Divergent according to the improper
integral exists or not.
Example
Evaluate
(a)
1

0
1
dx
x
(b)
1

1
1
dx
x3
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F. Ngai
Definite Integrals
Advanced Level Pure Mathematics

dx
x 9
Example
Evaluate

Example
Evaluate

Theorem
Let f (x) and g (x ) be two real-valued function continuous for x  a . If 0  f ( x)  g ( x)
0
2

dx
  x (1  x 2 )
2
x  a then the fact that


a
Example


a
f ( x)dx diverges implies
g ( x)dx converges implies that
Discuss whether


a


a
g ( x)dx diverges and the fact that
f ( x)dx converges.

dx
is convergence?
10 ln x

Prepared by K.
Page
25
F. Ngai
Definite Integrals
Advanced Level Pure Mathematics
Example
For any non-negative integer n , define

I n   x ne xdx .
0
(a) Show if n is positive integers, then I n 1  (n  1) I n .
Hence, if I n is convergent, I n 1 is also convergent.
(b) Find I n .
Prepared by K.
Page
26
F. Ngai