Explanation Given: Given improper integral is ∫ 0 ∞ x n e − x 2 d x . Formula used: For any a > 0 , ∫ a ∞ 1 x p d x converges for all p > 1 . ( Comparison Test )If f ( x ) ≥ g ( x ) ≥ 0 on theinterval [ a , ∞ ) then, (i) If ∫ a ∞ f ( x ) d x converges then ∫ a ∞ g ( x ) d x also converges. (xxvi) If ∫ a ∞ g ( x ) d x diverges then ∫ a ∞ f ( x ) d x also diverges. Calculation: ∫ 0 ∞ x n e − x 2 d x = ∫ 0 1 x n e − x 2 d x + ∫ 1 ∞ x n e − x 2 d x Since integral of every continuous function over a finite interval converges and x n e − x 2 is continuous, thus, ∫ 0 1 x n e − x 2 d x converges...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8, Problem 93CRE

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The improper integral $\int_{0}^{\infty} x^{n} e^{- x^{2}} d x$ converges for all $n > 0$ .

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Chapter 8, Problem 92CRE

Chapter 8, Problem 94CRE

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The improper integral ∫ 0 ∞ x n e − x 2 d x converges for all n > 0 .

Solution Summary: The author explains how the improper integral converges for all n > 0.

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The improper integral $\int_{0}^{\infty} x^{n} e^{- x^{2}} d x$ converges for all $n > 0$ .

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Chapter 8 Solutions

The improper integral ∫ 0 ∞ x n e − x 2 d x converges for all n &gt; 0 .

Solution Summary: The author explains how the improper integral converges for all n > 0.

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The improper integral ∫0∞xne−x2dxconverges for all n>0.

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The improper integral ∫ 0 ∞ x n e − x 2 d x converges for all n > 0 .

The improper integral $\int_{0}^{\infty} x^{n} e^{- x^{2}} d x$ converges for all $n > 0$ .