BC 3
Improper Integrals 3
Name:

Definition: An integral of the form
 f ( x) dx is called an improper integral. If
a



f ( x)  0 for x  a and lim   f ( x) dx  exists, then we say  f ( x ) dx converges and write:
b 
a
a

b



f ( x) dx  .

a f ( x) dx  lim

b 
a

b

Otherwise, we say
 f ( x) dx diverges.
a
Determine whether each of the following integrals converges or diverges. If it converges,
evaluate the integral.

1.

1

2.
1
dx
x
x
e
x2
dx
1
IMSA BC 3
Improper Integrals 3.1
Fall 14
There are other types of improper integrals all sharing the property that they represent the
area of an unbounded (but not necessarily of infinite area). Examples:
1
i.

0
1
dx ,
x
10
ii.

5

1
dx ,
x7
1
e
iii.

x2
dx ,
Sketch the graphs of the regions represented by each of these three integrals:
i.
ii.
iii.
In each of the above cases we can ask if the unbounded region has finite area. We frame
this questions as, “does the improper integral converge”?
We need to answer that question for these other types of unbounded areas.
So to that end: Determine with explanation whether each of the following integrals
converges with explanation:
1
1.

0
IMSA BC 3
1
dx
x

2.
1
 x dx

Improper Integrals 3.2
Fall 14
As we see below, improper integral that involve a region that is unbounded above are

 f ( x) dx .
defined in a manner similar to integrals of the type
a
Definition: If the function f is continuous and positive on the interval [a, b) and f has a
b
 f ( x) dx is called an improper integral. If
vertical asymptote at x  b , then
a


lim   f ( x) dx  exists, then we say
c b
a

b
b

a
b
 f ( x) dx converges and write:
a
b

f ( x) dx  lim   f ( x) dx  .
c b
a

b
Otherwise, we say
 f ( x) dx diverges.
a
b
In the above case, I say
 f ( x) dx has an impropriety at b. If f has a vertical asymptote at
a
x  c , with a  c  b , then again
b
 f ( x) dx is called an improper integral (with
a
b
impropriety at c). In this case,

a
c
b
a
c
f ( x) dx   f ( x) dx   f ( x ) dx and the integral
b
 f ( x) dx converges if and only if BOTH integrals on the right hand side converge.
a
1
Important: Infinite areas of opposite signs do not “cancel”. For example,
1
 x dx
1
diverges. The impropriety is at x  0 , so
1
1
0
1
1
1
 x dx   x dx   x dx .
1
1
0
1
Since both integrals on the right diverge (show this), the integral
1
 x dx
diverges.
1
Assignment: Read Section 7.7 and do exercises on pages 378-79: 4,5,12,16,19,26,34,35,40,41
IMSA BC 3
Improper Integrals 3.3
Fall 14