Explanation Given information: ∫ z ( ln z ) 2 d z Formula: ∫ u v d x = u ∫ v d x − ∫ d u d x ( ∫ v d x ) d x d f ( u ) d x = d f d u ⋅ d u d x ∫ x a d x = x a + 1 a + 1 + C d d x ( ln ( x ) ) = 1 x Calculation: Evaluate the integral. ∫ ( ln z ) 2 z d z = ( ln z ) 2 ∫ ( z ) d z − ∫ d ( ( ln z ) 2 ) d z ( ∫ ( z ) d z ) d z + C = ( ln z ) 2 ( z 2 2 ) − ∫ d ( ( ln z ) 2 ) d z ( z 2 2 ) d z + C Substitute u for ln z . ∫ ( ln z ) 2 z d z = u 2 ( z 2 2 ) − ∫ d ( u 2 ) d z ( z 2 2 ) d z + C Substitute f for u 2 . ∫ ( ln z ) 2 z d z = f ( z 2 2 ) − ∫ d f d z ( z 2 2 ) d z + C Apply the chain rule. ∫ ( ln z ) 2 z d z = f ( z 2 2 ) − ∫ ( d ( f ) d u ⋅ d u d z ) ( z 2 2 ) d z + C Substitute u 2 for f and ln z for u. ∫ ( ln z ) 2 z d z = u 2 ( z 2 2 ) − ∫ ( d ( u 2 ) d u ⋅ d ( ln z ) d z ) ( z 2 2 ) d z + C = u 2 ( z 2 2 ) − ∫ ( 2 u ( 1 z ) ) ( z 2 2 ) d z + C = u 2 ( z 2 2 ) − ∫ ( u z ) d z + C = u 2 ( z 2 2 ) − ∫ ( u z ) d z + C Substitute ln z for u

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Chapter 8.2, Problem 30E

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The integral by using a substitution prior to integration by parts.

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Chapter 8.2, Problem 29E

Chapter 8.2, Problem 31E

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