These are definite integrals where one or both of the following hold.
1. One of the limits is

or -

.
2. The integrand goes to

or -

for some finite value of x .
In each case we can relate it to a regular integral by using a limit. Let's look at some examples of the first case.
Example 1.
Evaluate
Ошибка!Ошибка!
dx .
Solution.
In a case like this we replace the upper limit by a variable which represents a finite number and then take the limit as that variable approaches infinity. So
Ошибка!Ошибка!
dx = Ошибка!
Ошибка!Ошибка!
dx
Now we evaluate the integral and take the limit as L approaches infinity. lim,
L
 
Ошибка!Ошибка!
dx = Ошибка!
tan -1 x Ошибка!
= Ошибка!
tan -1 L - tan -1 0 = lim,
L
 
tan -1 L - 0
= lim,
L
 
tan -1 L = Ошибка!
Example 2.
Evaluate Ошибка!Ошибка!
dx .
Solution.
This is similar. We replace the lower limit by a variable and then take the limit as that variable approaches minus infinity. So
Ошибка!Ошибка!
dx = Ошибка!
Ошибка!Ошибка!
dx = Ошибка!
tan -1 x
|
L
,
0
, = lim,
L

-

tan -1 0 - tan -1 L
= lim,
L

-

- tan -1 L = - (- Ошибка!
) = Ошибка!
If one limit is -

and the other is

, then we break the integral into two pieces and do each one separately.
Example 3.
Evaluate Ошибка!Ошибка!
dx .
Solution.
Ошибка!Ошибка!
dx = Ошибка!Ошибка!
dx + Ошибка!Ошибка!
dx =
Ошибка!
+ Ошибка!
=

2.5 - 1

Example 4.
Evaluate Ошибка!Ошибка!
dx .
Solution.
Ошибка!Ошибка!
dx = Ошибка!
Ошибка!Ошибка!
dx = Ошибка!
ln x Ошибка!
= lim,
L
 
ln L - ln 1 = lim,
L
 
ln L - 0
= lim,
L
 
ln L =

In this case the definite integral Ошибка!Ошибка!
dx = ln L approached infinity as L approaches infinity.
So we say the value of the integral is infinity and we also say the integral diverges . In the case where the integral is finite we say the integral converges .
Example 4.
Evaluate Ошибка!Ошибка!
dx .
Solution.
Ошибка!Ошибка!
dx = Ошибка!
Ошибка!x -2 dx = Ошибка!
Ошибка!
Ошибка!
= lim,
L
 
Ошибка!
Ошибка!
= Ошибка!
Ошибка!
- Ошибка!
= 1
Problem 1.
Show that Ошибка!Ошибка!
dx = Ошибка!
if p > 1.
Problem 2.
Find Ошибка!e -2 x dx , Ошибка!x e -2 x dx and Ошибка!x 2 e -2 x dx .
1.1 - 2