Explanation Given information: The integral function is ∫ − ∞ ∞ d x 4 x 2 + 4 x + 5 . Calculation: Show the integral function as follows: ∫ − ∞ ∞ d x 4 x 2 + 4 x + 5 (1) Show the definition of an Improper Integral of Type 1 : (a) If ∫ a t f ( x ) d x exists for every number t ≥ a , then ∫ a ∞ f ( x ) d x = lim t → ∞ ∫ a t f ( x ) d x Provided this limit exists (as a finite number). (b) If ∫ t b f ( x ) d x exists for every number t ≤ b , then ∫ − ∞ b f ( x ) d x = lim t → − ∞ ∫ t b f ( x ) d x Provided this limit exists (as a finite number). The integral functions ∫ a ∞ f ( x ) d x and ∫ − ∞ b f ( x ) d x are called convergent if the corresponding limit exists. The improper function ∫ a ∞ f ( x ) d x and ∫ − ∞ b f ( x ) d x are called divergent if the corresponding limit does not exist. (c) If both the integral function ∫ a ∞ f ( x ) d x and ∫ − ∞ a f ( x ) d x are convergent, then ∫ − ∞ ∞ f ( x ) d x = ∫ − ∞ a f ( x ) d x + ∫ a ∞ f ( x ) d x Here, a is any real number. Apply Part (c) of the definition of improper integral in Equation (1). ∫ − ∞ ∞ d x 4 x 2 + 4 x + 5 = lim t → − ∞ ∫ t ∞ d x 4 x 2 + 4 x + 5 (2) Consider the value of the function u = ( 2 x + 1 ) (3) Differentiate Equation (3) with respect to x . d u d x = d d x ( 2 x + 1 ) d u d x = 2 d u = 2 d x Modify Equation (2). Substitute u for ( 2 x + 1 ) and d u 2 for d x in Equation (1)

Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

2nd Edition

ISBN: 9781285101552

Author: James Stewart

Publisher: Cenage

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Chapter 6, Problem 49RE

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To evaluate: The integral function if it is convergent otherwise to show the integral function as divergent.

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Chapter 6, Problem 48RE

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Essential Calculus: Early Transcendentals; MAC 2311 Sequence| MAC 2281Sequence USF (Essential Calculus)

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The integral function if it is convergent otherwise to show the integral function as divergent.

Solution Summary: The author evaluates the integral function if it is convergent, otherwise to show it as divergent.

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To evaluate: The integral function if it is convergent otherwise to show the integral function as divergent.

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Chapter 6 Solutions