   Chapter 8.8, Problem 52E

Chapter
Section
Textbook Problem

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 ∫ a ∞ f ( x ) dx converges, then ∫ a ∞ g ( x ) dx also converges, and if ∫ a ∞ g ( x ) dx diverges, then ∫ a ∞ f ( x ) dx also diverges. This is known as the Comparison Test for improper integrals.(a) Use the Comparison Test to determine whether ∫ 1 ∞ e − x 2 dx converges or diverges. (Hint: Use the fact that e − x 2 ≤ e − x  for  x   ≥  1 . )(b) Use the Comparison Test to determine whether ∫ 1 ∞ 1 x 5 + 1 dx converges or diverges. (Hint: Use the fact that 1 x 5 + 1 ≤ 1 x 5  for  x   ≥  1 .)

(a)

To determine
Whether the improper integral 1ex2dx converges or not.

Explanation

According to the comparison test,

Suppose the functions f and g are continuous on the interval [a,) and 0g(x)f(x).

If af(x)dx converges, then ag(x)dx also converges, and if af(x)dx diverges, then ag(x)dx also diverges.

Consider the provided integral,

1ex2dx

Since ex2ex on the interval [1,).

So, for the convergence of 1exdx, the integral 1ex2dx also converges according to the comparison test

(b)

To determine
Whether the improper integral 11x5+1dx converges or not.

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 