1. 7.8 Improper Integrals
Definition:
The definite integral is called an improper integral if
At least one of the limits of integration is infinite, or
The integrand f(x) has one or more points of discontinuity on the interval [a, b].

2. Infinite Limits of Integration
If f is continuous on the interval , then
If f is continuous on the interval , then
If f is continuous on the interval , then
where c is any real number
In each case, if the limit exists, then the improper integral is said to converge; otherwise, the improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.

3. Discontinuous Integrand
If f is continuous on the interval and has an infinite discontinuity at b, then
If f is continuous on the interval and has an infinite discontinuity at a, then
If f is continuous on the interval except for some c in (a, b) at which f has an infinite discontinuity, then
In each case, if the limit exists, then the improper integral is said to converge; otherwise, the improper integral diverges. In the third case, the improper integral on the left diverges if either of the improper integrals on the right diverges.

4. Examples:
Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1)
Now,

5. 2)
D I
Note:
Thus,

6. 3) 4)
Now,
where

7. Comparison Theorem Suppose that f and g are continuous functions with
We use the Comparison Theorem at times when its impossible to find the exact value of an improper integral.
Suppose that f and g are continuous functions with
for
If is convergent, then
is convergent,
(b) If is divergent, then
is divergent

8. Examples:
Use the Comparison Theorem to determine whether the integral is convergent or divergent. 1)
Observe that
Now,
Note:
Thus,

9. 2)
Observe that
Now,