7.8 Improper Integrals

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 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,