Explanation Definition used: Improper integral “If f is continuous over the indicated interval and the limit exists, then 1. ∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x 2. ∫ − ∞ b f ( x ) d x = lim a → − ∞ ∫ a b f ( x ) d x 3. ∫ − ∞ ∞ f ( x ) d x = ∫ − ∞ c f ( x ) d x + ∫ c ∞ f ( x ) d x where c is any point on ( − ∞ , ∞ ) , provided both improper integrals on the right exist”. 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 ∫ − ∞ ∞ x 1 + x 2 d x . The integrand f ( x ) = x 1 + x 2 is finite on the unbounded interval ( − ∞ , ∞ ) . Use the definition and theorem mentioned above to check convergence of the improper integral. ∫ − ∞ ∞ x 1 + x 2 d x = lim a → − ∞ ∫ a c x 1 + x 2 d x + lim b → ∞ ∫ c b x 1 + x 2 d x = 1 2 lim a → − ∞ ∫ a c 1 1 + x 2 d ( 1 + x 2 ) + 1 2 lim b → ∞ ∫

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 53E

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To find: The value of improper integral ∫−∞∞x1+x2dx, if it converges.

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Chapter 11.1, Problem 52E

Chapter 11.1, Problem 54E

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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 ∫ − ∞ ∞ x 1 + x 2 d x , if it converges.

Solution Summary: The author explains that the value of improper integral displaystyle 'underset-infty', if it converges, diverges. If f is continuous over the indicated interval and

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To find: The value of improper integral ∫−∞∞x1+x2dx, if it converges.

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Chapter 11 Solutions