Explanation Given Information: The provided integral is, ∫ − ∞ ∞ x 3 e − x 4 d x Formula used: According to the exponential rule of integrals, ∫ e x d x = e x + C If the limit defining the improper integral is a unique finite number, then integral converges else it diverges, ∫ − ∞ ∞ f ( x ) d x = lim a → − ∞ ∫ a c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x Calculation: Consider the provided integral, ∫ − ∞ ∞ x 3 e − x 4 d x Now, use the formula, ∫ − ∞ ∞ f ( x ) d x = lim a → − ∞ ∫ a c f ( x ) d x + lim b → ∞ ∫ c b f ( x ) d x Multiply and divide by − 4 and rewrite the integral as, ∫ − ∞ ∞ x 3 e − x 4 d x = − 1 4 [ lim a → − ∞ ∫ a 0 − 4 x 3 e − x 4 d x + lim b → ∞ ∫ 0 b − 4 x 3 e − x 4 d x ] Let − x 4 = t , then differentiate both the sides − 4 x 3 d x = d t Thus, the integral becomes, ∫ − ∞ ∞ x 3 e − x 4 d x = − 1 4 [ lim a → − ∞ ∫ a 0 e t d t + lim b → ∞ ∫ 0 b e t d t ]

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 13.7, Problem 19E

To determine

To calculate: The value of the improper integral ∫−∞∞x3e−x4dx if it converges.

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Chapter 13.7, Problem 18E

Chapter 13.7, Problem 20E

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Chapter 13 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 13.1 - CHECKPOINT 1. For the interval [0, 2]. determine...Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

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The value of the improper integral ∫ − ∞ ∞ x 3 e − x 4 d x if it converges.

Solution Summary: The author calculates the value of the provided integral if it converges. If the limit defining the improper integral is a unique finite number, it diverges.

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To calculate: The value of the improper integral ∫−∞∞x3e−x4dx if it converges.

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Chapter 13 Solutions