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Unit 2
...

x

Area  



1

1
dx
2
x

Area  

5

2

1
dx
x2

Improper Integral

An integral is improper if either limits of
integration are infinite or if the integrand has
infinite discontinuity at a number or at
numbers within the interval of integration
...
If f is continuous for all x  a, then





a

f  x  dx  lim  f  x  dx
...
If f is continuous for all x  b, then



b



f  x  dx  lim  f  x  dx
...
If f is continuous for all values of x and
c is any real number, then







c





c

f  x  dx   f  x  dx  

f  x  dx

 lim  f  x  dx  lim  f  x  dx
c

t  t

t

t  c

Example 2
...
1
Evaluate 



1





1

dx

...

Otherwise, it is said to be divergent
...
7
...

1
...




dx
2
...
If f is continuous for all x in  a, b and if
f has an infinite discontinuity at a then

 f  x  dx  lim  f  x  dx
b

a

b

t  a

t

2
...
If f is continuous for all x in  a, b except
at c where a  x  c and if f has an
infinite discontinuity at c then

 f  x  dx   f  x  dx   f  x  dx
 lim  f  x  dx  lim  f  x  dx
b

c

b

a

a

c

t

t c

a

b

t c

t

Example 2
...
3
Evaluate the integrals
...

2
x2
5
dx
 lim 
t  2 t
x2
 lim

t 2



3

t 2

u

 12
3

du

 lim 2u 2 
t  2
t  2 


1

Let u  x  2
du  dx
x t u t 2
x 5u 3

3

 lim 2u 2 
t  2
t  2 


1



 lim 2 3  2  t  2 

t 2

2 3

1

2



1 dx
3 dx
dx
2
...

1

dx
Since, 
is divergent,
0 x 1
3 dx
0 x  1 is also divergent
...

Exercises 7
...
10 (1-29) on page 664
Check your answers on A-161 and A-162
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7



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