(a) Explanation According to the comparison test, Suppose the functions f and g are continuous on the interval [ a , ∞ ) and 0 ≤ g ( x ) ≤ f ( x ) . If ∫ a ∞ f ( x ) d x converges, then ∫ a ∞ g ( x ) d x also converges, and if ∫ a ∞ f ( x ) d x diverges, then ∫ a ∞ g ( x ) d x also diverges. Consider the provided integral, ∫ 1 ∞ e − x 2 d x Since e − x 2 ≤ e − x on the interval [ 1 , ∞ ) . So, for the convergence of ∫ 1 ∞ e − x d x , the integral ∫ 1 ∞ e − x 2 d x also converges according to the comparison test (b) To determine Whether the improper integral ∫ 1 ∞ 1 x 5 + 1 d x converges or not.

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Author: Ron Larson, Bruce H. Edwards

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Question

Chapter 8.8, Problem 52E

(a)

To determine

Whether the improper integral ∫1∞e−x2dx converges or not.

(b)

To determine

Whether the improper integral ∫1∞1x5+1dx converges or not.

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Chapter 8.8, Problem 51E

Chapter 8.8, Problem 53E

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Whether the improper integral ∫ 1 ∞ e − x 2 d x converges or not.

Solution Summary: The author analyzes whether the improper integral displaystyle 'int' converges or not according to the comparison test.