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IMPROPER INTEGRALS
Section 9-0
An integral is improper if:
a. One or both limits of integration are infinite
b. The function has an infinite discontinuity (a
vertical asymptote) at or between the limits
c. Both a and b hold
a  x  b or [a, b]
Integration over a finite interval
a  x   or [a, ]
Integration over an infinite interval
Explain why each of the following are improper

1. 1 dx
1 x
2.

1
 x 2  1 dx
Explain why each of the following are improper
5
3.

1
1
dx
x 1
2
4.
1
2 x  12 dx
The Fundamental Theorem of Calculus
States that, if f is continuous on the closed
interval [a, b] and f is the antidervative then
b
 f ' ( x)dx  f (a)  f (b)
a
Note: only applies to closed intervals and
improper integrals are unbounded intervals
Change the Form
We will integrate over some
finite interval [a, b] so we can
use the rules for definite
integrals then evaluate what
happens as b becomes very
large.

b
 f ( x)dx  lim  f ( x)dx
a
b
a
1. evaluate the integral for a constant (b)
2. take the limit of the resulting integral as (b)
approaches the limit in question, usually ±∞
Converges or Diverges
1. If the limit exists, the
integral converges then
the value of the integral is
the limit. In this case we
say that the improper
integral is convergent
2. If the limit does not exist,
or it is infinite; meaning gets
infinitely large without
bound, then we say that the
improper integral is
divergent
5) Evaluate

1
4 x  23 dx
6) Evaluate the improper integral and identify
if it converges or diverges

1
1 x 2 dx

y
x




7) Evaluate the improper integral and identify
if it converges or diverges

1
1 x dx

y
x




8) Evaluate the improper integral and identify
if it converges or diverges

 cos xdx
0
9) Evaluate the improper integral and identify
if it converges or diverges

x
xe
 dx
1
Vertical Asymptotes
10) Evaluate the improper integral and identify
if it converges or diverges
2
1
1 x 3 dx
HOME WORK
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