Explanation Given: The definite integral ∫ 1 / 8 e / 8 ln ( 8 x ) d x . Formula used: Write the formula for integral of f ( x ) = x n provided n ≠ − 1 . ∫ x n d x = x n + 1 n + 1 + C Write the formula for integral of f ( x ) = k , k being a real number. ∫ k d x = k x + C In the above formulae, C is the integration constant. The use of integration constant C is done only for indefinite integrals. Write the formula for derivative of f ( x ) = ln x . d d x ( ln x ) = 1 x The integration by parts for functions u and v is given by the formula: ∫ u d v = u v − ∫ v d u Here, d u and d v indicate the derivatives of functions u and v respectively. The definite integral of a function f ′ ( x ) from x = a to x = b is calculated by the formula: ∫ a b f ′ ( x ) d x = | f ( x ) | a b = f ( b ) − f ( a ) Calculation: Consider the given definite integral: I = ∫ 1 / 8 e / 8 ln ( 8 x ) d x Use the integration by parts formula to solve for I where u is ln ( 8 x ) and dv is 1 d x . Therefore, d u = 1 8 x d x and v = x . Write the integration by parts formula. ∫ u d v = u v − ∫ v d u Substitute ln ( 8 x ) for u , 1 d x for dv, ( 1 8 x ) 8 d x for d u and x for v

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ISBN: 8220101336323

Author: MULLINS

Publisher: PEARSON

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Chapter 13, Problem 57RE

To determine

To calculate: The value of the definite integral given below:

∫1/8e/8ln(8x)dx

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Chapter 13, Problem 35RE

Chapter 13, Problem 58RE

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