Explanation Given Information: The provided indefinite integral is ∫ x ( ln x ) 2 d x . Formula used: The integration by parts ∫ u d v = u v − ∫ v d u . Where, u and v are function of x . ∫ x n d x = x n + 1 n + 1 , n ≠ − 1 Calculation: Consider the indefinite integral ∫ x ( ln x ) 2 d x Here, d v = x d x and u = ( ln x ) 2 First find v , d v = x d x ∫ d v = ∫ x d x On further solving, v = x 2 2 …...…... (1) Now find d u , u = ( ln x ) 2 Differentiate both side with respect x , d u d x = d ( ( ln x ) 2 ) d x d u d x = 2 ln x x And, d u = 2 ln x x ⋅ d x …...…... (2) Apply integration by parts formula and substitute equation (1) and (2) in ∫ u d v = u v − ∫ v d u , ∫ x ( ln x ) 2 d x = x 2 2 ( ln x ) 2 − ∫ x ln x d x Use integration by parts on the second integral ∫ x ln x d x Here, d v = x d x and u = ln x First find v , d v = x d x ∫ d v = ∫ x d x On further solving, v = x 2 2 …...…... (3) Now find d u , u = ln x Differentiate both side with respect x , d u d x = d ( ln x ) d x d u d x = 1 x And, d u = 1 x ⋅ d x …

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.1, Problem 27E

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To calculate: The value of indefinite integral ∫x(lnx)2dx.

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Chapter 6.1, Problem 26E

Chapter 6.1, Problem 28E

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Chapter 6 Solutions

Calculus: An Applied Approach (MindTap Course List)

Ch. 6.1 - Prob. 1CP Ch. 6.1 - Prob. 2CP Ch. 6.1 - Prob. 3CP Ch. 6.1 - Prob. 4CP Ch. 6.1 - Prob. 5CP Ch. 6.1 - Prob. 6CP Ch. 6.1 - Prob. 7CP Ch. 6.1 - Prob. 1SWU Ch. 6.1 - Prob. 2SWU Ch. 6.1 - Prob. 3SWU

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The value of indefinite integral ∫ x ( ln x ) 2 d x .

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To calculate: The value of indefinite integral ∫x(lnx)2dx.

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Chapter 6 Solutions