Explanation Given information : ∫ 3 ∞ d x 2 x − 1 3 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 Rewrite the integral by changing the radical to rational exponent ∫ 3 ∞ d x 2 x − 1 3 = ∫ 3 ∞ d x ( 2 x − 1 ) 1 / 3 Convert the integral to a limit ∫ 3 ∞ d x ( 2 x − 1 ) 1 3 = lim t → ∞ ∫ 3 t d x ( 2 x − 1 ) 1 3 Using the u-substitution to rewrite the integral L e t u = 2 x − 1 ; d u = 2 ⋅ d x 2 ⋅ d x = d u d x = d u 2 lim t → ∞ ∫ 3 t d x ( 2 x − 1 ) 1 3 = lim t → ∞ ∫ 3 t d u 2 u 1 3 = 1 2 lim t → ∞ ∫ 3 t d u u 1 3 Finding the integral and rewriting integral in terms of x 1 2 lim t → ∞ [ u 1 3 + 1 1 3 + 1 ] 3 t = 1 2 lim t → ∞ [ u 1 3 + 3 3 1 3 + 3 3 ] 3 t x

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 7.7, Problem 10PS

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Chapter 7.7, Problem 9PS

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Chapter 7 Solutions

Calculus

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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