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 ∫ 1 ∞ ln x x d x , where p is positive constant. The integrand f ( x ) = ln x x is finite on the unbounded interval [ 1 , ∞ ) . Use the definition and theorem mentioned above to check convergence of the improper integral. ∫ 1 ∞ ln x x d x = lim b → ∞ ∫ 1 b ln x x d x = lim b → ∞ ∫ 1 b ln ( x ) d ( ln x )

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 11.1, Problem 59E

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To find: The value of improper integral ∫1∞lnxxdx, if it converges.

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Chapter 11.1, Problem 58E

Chapter 11.1, Problem 60E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

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 ∫ 1 ∞ ln x 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 ∫1∞lnxxdx, if it converges.

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Chapter 11 Solutions