Explanation Definition used: Improper integral Let a function f is continuous on the unbounded interval [ a , ∞ ) . Then the improper integral of f over [ a , ∞ ) is defined as ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x . If it exists and is finite, then it is convergent otherwise it is divergent. Theorem used: Fundamental Theorem “Let f be a continuous function on [ a , b ] . If F is anti-derivative of f on [ a , b ] , then ∫ a b f ( x ) d x = F ( b ) − F ( a ) ”. Calculation: The given improper integral is ∫ 0 ∞ ( e − x + 2 e x ) d x , where k is constant. The integrand f ( x ) = e − x + 2 e x is finite on the unbounded interval [ 0 , ∞ ) . Use the definition and theorem mentioned above to check convergence of the improper integral. ∫ 0 ∞ ( e − x + 2 e x ) d x = lim b → ∞ ∫ 0 b ( e − x + 2 e x ) d x = lim b → ∞ ∫ 0 b e − x d x + 2 lim b → ∞ ∫ 0 b e x d x = − lim b → ∞ [ e − x ] 0 b + 2

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

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Chapter 11.1, Problem 56E

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To find: The value of improper integral ∫0∞(e−x+2ex)dx, if it converges.

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Chapter 11.1, Problem 55E

Chapter 11.1, Problem 57E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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The value of improper integral ∫ 0 ∞ ( e − x + 2 e x ) d x , if it converges.

Solution Summary: The author calculates the value of the improper integral, if it converges, otherwise it diverges.

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To find: The value of improper integral ∫0∞(e−x+2ex)dx, if it converges.

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Chapter 11 Solutions