Explanation Given information: ∫ 0 π 2 x sin 8 x d x Formula Used: Integration by parts rule: The integrand is the product of two functions f ( x ) and g ( x ) , then ∫ f ( x ) g ( x ) = f ( x ) G ( x ) − ∫ f ' ( x ) G ( x ) d x , where G ( x ) is the antiderivative of g ( x ) and f ' ( x ) is a derivative of f ( x ) . Calculation: Let f ( x ) = x and g ( x ) = sin 8 x . Differentiate f ( x ) = x with respect to x . ⇒ f ' ( x ) = 1 Integrate g ( x ) = sin 8 x with respect to x . G ( x ) = ∫ g ( x ) d x = ∫ sin 8 x d x = − 1 8 cos 8 x By using the integration by parts rule, ∫ x sin 8 x d x = x ( − 1 8 cos 8 x ) − ∫ 1 ( − 1 8 cos 8 x ) d x = − 1 8 x cos 8 x + 1 8 ∫ cos 8 x d x = − 1 8 x cos 8 x + 1 8 ( 1 8 sin 8 x ) + C , where C is any arbitrary constant

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ISBN: 9780137590469

Author: Larry Goldstein / David Lay

Publisher: Pearson Education (US)

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Chapter 9, Problem 38RE

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To calculate: ∫0π2xsin 8x dx.

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Chapter 9, Problem 37RE

Chapter 9, Problem 39RE

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∫ 0 π 2 x sin 8 x d x .

Solution Summary: The author explains the formula used for integration by parts rule: the integrand is the product of two functions f(x) and

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To calculate: ∫0π2xsin 8x dx.

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Chapter 9 Solutions