Explanation Formula used: d d x ( ln x ) = 1 x (1) d d x ( C ) = 0 , where C is a constant (2) Chain rule: d d x f [ g ( x ) ] = f ′ [ g ( x ) ] g ′ ( x ) Calculation: The given formula is ∫ 1 x d x = ln | x | + C . Integration and differentiation are inverse operations. So, the differentiation of ∫ f ( x ) d x is the integrand f ( x ) . Notice that the function ln | x | is defined as, ln | x | = { ln ( x ) , x > 0 ln ( − x ) , x < 0 } Here, f ( x ) = 1 x . ∫ f ( x ) d x = ln | x | + C = ln ( − x ) + C , for x < 0 Use formula (1) and (2) to differentiate the integral ∫ f ( x ) d x as below

COLL.MATH F/BUS ECO.(LL+18 WEEK MYLAB E)

14th Edition

ISBN: 9780135998083

Author: Barnett

Publisher: PEARSON

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Chapter 12.1, Problem 78E

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To justify: The formula ∫1xdx=ln|x|+C by using differentiation for x<0.

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Chapter 12.1, Problem 77E

Chapter 12.1, Problem 79E

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COLL.MATH F/BUS ECO.(LL+18 WEEK MYLAB E)

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To justify: The formula ∫ 1 x d x = ln | x | + C by using differentiation for x &lt; 0 .

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To justify: The formula ∫1xdx=ln|x|+C by using differentiation for x<0.

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Chapter 12 Solutions

To justify: The formula ∫ 1 x d x = ln | x | + C by using differentiation for x < 0 .