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

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.

Example 1 Evaluate the following:

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.

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

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

Example: Determine whether converges or diverges by comparing it with