1. 10.4 Integrals with Discontinuous Integrands. Integral comparison test.
Rita Korsunsky

2. Integrals with Discontinuous Integrands are also Improper Integrals
Definition: a b
i) If f is continuous on [a, b) and discontinuous
at b, then
ii) If f is continuous on (a, b] and discontinuous
at a, then
iii) if f has discontinuity at a number c in the
interval [a,b], then
provided the limit exists.

3. Example 1
Evaluate the following:

4. Example 2
Determine whether the following improper integral converges or diverges.
First evaluate this:
(diverges)
(diverges)
If one of the integrals in the sum diverges, we can conclude that the entire integral diverges.

5. Example 3 Evaluate the following: Similarly: First evaluate this:
Add both values:

6. Integral Comparison Test
Suppose f and g are continuous and
for every x in (a,b]. If f and g are discontinuous at x=a, then the following comparison tests can be proved:
a=0
b=1

7. Example:
Determine whether converges or diverges
by comparing it with