Explanation Given information: ∫ 1 2 x − 3 / 2 ln x d x Formula Used: Integration by parts rule: The integrand is the product of two functions f ( x ) and g ( x ) , then ∫ f ( x ) g ( x ) = f ( x ) G ( x ) − ∫ f ' ( x ) G ( x ) d x , where G ( x ) is the antiderivative of g ( x ) and f ' ( x ) is a derivative of f ( x ) . Calculation: Let f ( x ) = ln x and g ( x ) = x − 3 / 2 . Differentiate f ( x ) = ln x with respect to x . ⇒ f ' ( x ) = 1 x Integrate g ( x ) = x − 3 / 2 with respect to x . G ( x ) = ∫ g ( x ) d x = ∫ x − 3 / 2 d x = x − 1 / 2 − 1 / 2 = − 2 x − 1 / 2 By using the integration by parts rule, ∫ x − 3 / 2 ln x d x = ln x ( − 2 x − 1 / 2 ) − ∫ 1 x ( − 2 x − 1 / 2 ) d x = − 2 x − 1 / 2 ln x + 2 ∫ x − 3 / 2 d x = − 2 x − 1 / 2 ln x + 2 ( x − 1 / 2 − 1 / 2 ) + C , where C is an arbitrary constant

12th Edition

ISBN: 9780137590810

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar, William Edward Tavernetti

Publisher: PEARSON

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Chapter 9, Problem 42RE

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To calculate: ∫12x−3/2 lnx dx.

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Chapter 9, Problem 41RE

Chapter 9, Problem 43RE

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Chapter 9 Solutions

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∫ 1 2 x − 3 / 2 ln x d x .

Solution Summary: The author explains how to calculate the integrand, which is the product of two functions.

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To calculate: ∫12x−3/2 lnx dx.

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Chapter 9 Solutions