Comparison Test for Improper Integrals In some cases, it is impossible to find the exact value of an improper integral, but it is important to determine whether the integral converges or diverges. Suppose the functions f and g are continuous and 0 ≤ g(x) ≤ f(x) on the interval [a,). It can be shown that if
dx converges, then
dx also converges, and if
dx diverges, then
dx also diverges. This is known as the Comparison Test for improper integrals.
(a) Use the Comparison Test to determine whether
dx converges or diverges. (Hint: Use the fact that )
(b) Use the Comparison Test to determine whether
dx converges or diverges. (Hint: Use the fact that
