To determine To find: The value of the improper integral ∫ − ∞ ∞ x e − x 2 d x . Explanation Definition used: 1. “If f is continuous on ( − ∞ , c ] , then ∫ − ∞ c f ( x ) d x = lim b → − ∞ ∫ b c f ( x ) d x .” 2. “If f is continuous on [ c , ∞ ) , then ∫ c ∞ f ( x ) d x = lim b → ∞ ∫ c b f ( x ) d x .” 3. Let a function f be continuous over the unbounded interval ( − ∞ , ∞ ) . Let c be any real number, and suppose that both the improper integral ∫ − ∞ c f ( x ) d x and ∫ c ∞ f ( x ) d x are convergent. Then the improper integral of f over ( − ∞ , ∞ ) is defined as ∫ − ∞ ∞ f ( x ) d x = ∫ − ∞ c f ( x ) d x + ∫ c ∞ f ( x ) d x , If anyone of the right hand side is divergent then ∫ − ∞ ∞ f ( x ) d x is divergent. Calculation: The improper integral is ∫ − ∞ ∞ x e − x 2 d x . In order to find the above integral, first evaluate the ∫ − ∞ 0 x e − x 2 d x and ∫ 0 ∞ x e − x 2 d x . Evaluate the integral ∫ − ∞ 0 x e − x 2 d x as follows. ∫ − ∞ 0 x e − x 2 d x = lim b → − ∞ ∫ b 0 x e − x 2 d x = − lim b → − ∞ 1 2 ∫ b 0 e − x 2 d ( x 2 ) = − 1 2 lim b → − ∞ [ e − x 2 ] b 0 = − 1 2 lim b → − ∞ ( e − 0 2 ) + 1 2 lim b → − ∞ ( e − b 2 ) On further simplification, ∫ − ∞ 0

Calculus with Applications (11th Edition)

11th Edition

ISBN: 9780321979421

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Question

Chapter 8.4, Problem 35E

To determine

To find: The value of the improper integral ∫−∞∞xe−x2dx .

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Chapter 8.4, Problem 34E

Chapter 8.4, Problem 36E

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Chapter 8 Solutions

Calculus with Applications (11th Edition)

Ch. 8.1 - YOUR TURN 1 Find Ch. 8.1 - YOUR TURN 2 Find Ch. 8.1 - YOUR TURN 3 Find Ch. 8.1 - YOUR TURN 4 Find . Ch. 8.1 - YOUR TURN 5 Find Ch. 8.1 - Prob. 1WE Ch. 8.1 - Find the following. W2. Ch. 8.1 - Find the following. W3. Ch. 8.1 - Find the following. W4. Ch. 8.1 - Use integration by parts to find the integrals in...

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