Explanation Given information: ln | x | Formula: ∫ a b 1 x n d x = [ x − n + 1 − n + 1 ] a b d d x ( ln x ) = [ 1 x ] d d x ( ln ( − x ) ) = [ 1 x ] ∫ u ( x ) v ′ ( x ) d x = u ( x ) v ( x ) − ∫ v ( x ) u ′ ( x ) d x (1) Calculation: Test the below expression using the direct comparison test. ln | x | Write the function f ( x ) as below. f ( x ) = ln | x | Write the function g ( x ) as below. g ( x ) = 1 x The following term cannot be tested using this direct comparison test as the following condition exists as given below. f ( x ) > g ( x ) on [ − 1 , 0 ] f ( x ) < g ( x ) on [ 0 , 1 ] This does not satisfy the condition of continuity within the given limit [ − 1 , 1 ] and the condition 0 ≤ f ( x ) ≤ g ( x ) . The discontinuity exists in the function at the value of 0 for x , where it will reach infinity. Hence, use the limit comparison test. Hence, use the limit comparison test. Test the below expression using the limit comparison test. ln | x | Apply also the following test for the convergence as below. lim x → ∞ f ( x ) g ( x ) = lim x → ∞ ln | x | 1 x = lim x → ∞ x ln | x | = ∞ Therefore, the term is not within the range ( 0 , ∞ ) . Hence, the integral term f ( x ) cannot be tested with this limit comparison test. Express the integral f ( x ) as below. ∫ − 1 1 ln | x | d x = ∫ − 1 0 ln ( − x ) d x + ∫ 0 1 ln ( x ) d x (2) Consider the term ∫ 0 1 ln ( x ) d x from the equation (2). Let us take the term ln x as u ( x ) . u ( x ) = ln x Differentiate the term u with respect to x . u ′ ( x ) = 1 x d x Let us take the term 1 as v ′ ( x ) . Integrate the term with respect to x . ∫ v ′ ( x ) = ∫ d x v ( x ) = x Substitute ln x for u ( x ) , 1 x d x for u ′ ( x ) , x for v ( x ) and 1 for v ′ ( x ) in equation (1)

4th Edition

ISBN: 9780134439099

Author: Hass, Joel., Heil, Christopher , WEIR, Maurice D.

Publisher: Pearson,

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Chapter 8.8, Problem 49E

To determine

Whether an integral converges or diverges by testing it either using the direct comparison test, or the Limit comparison test by adopting any one of the appropriate method.

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Chapter 8.8, Problem 48E

Chapter 8.8, Problem 50E

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