1. Section 7.7 Improper Integrals

2. So far in computing definite integrals on the interval a ≤ x ≤ b, we have assumed our interval was of finite length and our function was continuous This is not always the case We have to be able to compute integrals that are on unbounded intervals We need to be able to compute integrals of functions that may not be continuous on our given interval Such integrals are called improper

3. I. When the limit of integration is infinite Consider We calculate Now we take the limit as b ∞ So we say converges to 1

4. Suppose f(x) is positive for x ≥ a. If is a definite number, we say that converges. Otherwise it diverges.

5. Examples

6. Recall If either of the integrals diverges, the whole thing diverges

7. II. When the integrand becomes infinite In this case we may have a finite interval, but the function may be unbounded somewhere on the interval Consider Has an asymptote at x = 0 Handle it in a similar way

8. We compute Now we take the limit So converges to 2

9. Suppose f(x) is positive for x ≥ a. If is a definite number, we say that converges. Otherwise it diverges. As before we can split the interval so the point of discontinuity is an end point

10. Examples