EXPLORING CONCEPTS
Improper Integral Consider the integral
To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals for the given integral to converge?
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Writing
(a) The improper integrals
diverge and converge, respectively. Describe the essential difference between the integrands that cause one integral to converge and the other to diverge.
(b) Use a graphing utility to graph the function over the interval . Use your knowledge of the definite integral to make an inference as to whether the integral
converges. Give reasons for your answer.
(c) Use one application of integration by parts and the result of Exercise 89 to determine the divergence or convergence of the integral in part (b).
Laplace Transforms Let be a function defined for all positive values of t. The Laplace Transform of is defined by
Comparison Test for Improper Integrals In some cases, it is impossible to find the exact value of improper integral, but it is important to determine whether the integral converges or diverges. Suppose the functions f and g are continuous and on the interval It can be shown that if
dx converges, then
dx alos converges if
dx diverges, then
dx alos diverges. This is know as the Comparison Test for improper integrals.
(a) Use the comparison Test to determine whether
dx converges or diverges. (Hint: Use the fact that
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 also converges, and if
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
