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].

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.

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.

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

2) D I Note: Thus,

3) 4) Now, where

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

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

2) Observe that Now,