Explanation Given information : ∫ − ∞ 0 d x 2 − x Formula Used: a m n = a m / n Negative exponent rule: 1 a m = a − m Power rule of integral: ∫ a b x n ⋅ d x = [ x n + 1 n + 1 ] a b Concept Involved: If ∫ a t f ( x ) ⋅ d x exists for every t > a , then ∫ a ∞ f ( x ) ⋅ d x = lim t → ∞ ∫ a t f ( x ) ⋅ d x provided the limit exists and is finite. If ∫ t b f ( x ) ⋅ d x exists for every t < b , then ∫ − ∞ b f ( x ) ⋅ d x = lim t → − ∞ ∫ t b f ( x ) ⋅ d x provided the limit exists and is finite. If a > 0 then ∫ a ∞ 1 x p d x is convergent if p > 1 and divergent if p ≤ 1 Calculation: Description Steps Convert the integral to a limit ∫ − ∞ 0 d x 2 − x = lim t → − ∞ ∫ t 0 d x 2 − x Using the u-substitution to rewrite the integral L e t u = 2 − x ; d u = − d x − d x = d u d x = − d u lim t → − ∞ ∫ t 0 d x 2 − x = − lim t → − ∞ ∫ t 0 d u u Rewrite the integral by changing the radical to rational exponent − lim t → − ∞ ∫ t 0 d u u 1 / 2 Using the negative exponent rule rewrite the integral − lim t → − ∞ ∫ t 0 u − 1 / 2 d u Finding the integral and rewriting integral in terms of x − lim t → − ∞ [ u − 1 2 + 1 − 1 2 + 1

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

6th Edition

ISBN: 9781524908102

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 7.7, Problem 25PS

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Chapter 7 Solutions

Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)

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Tofind : the value of improper integral if it converges or show that it diverges.

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Chapter 7 Solutions