Explanation Given: The formula is: ∫ cosh n x d x = 1 n cosh n − 1 x sinh x + n − 1 n ∫ cosh n − 2 x d x . Formula used: Steps of integration by parts: ∫ u v d x = u ∫ v d x − ∫ u ′ ( ∫ v d x ) d x Integration formula’s: ∫ cosh x d x = sinh x + c Calculation: The formula is: ∫ cosh n x d x = 1 n cosh n − 1 x sinh x + n − 1 n ∫ cosh n − 2 x d x . The intgeral is ∫ cosh n x d x . Let I n = ∫ cosh n x d x We can rewrite it as: I n = ∫ cosh n x d x = ∫ cosh n − 1 x cosh x d x Integrate the function by the formula of integrating by parts. I n = ∫ cosh n − 1 x cosh x d x = cosh n − 1 x ∫ cosh x d x − ∫ [ d d x ( cosh n − 1 x ) ⋅ ∫ cosh x d x ] d x = cosh n − 1 x ⋅ sinh x − ( n − 1 ) ∫ [ ( cosh n − 2 x ) sinh x ⋅ sinh x ] d x = cosh n − 1 x ⋅ sinh x − ( n − 1 ) ∫ [ ( cosh n − 2 x ) sinh 2 x ] d x Use the formula: sinh 2 x = cosh 2 x − 1 I n = cosh n − 1 x ⋅ sinh x − ( n −

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Chapter 8.4, Problem 35E

To determine

Prove the reduction formula for $n \geq 2$ .

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Chapter 8.4, Problem 34E

Chapter 8.4, Problem 36E

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Chapter 8 Solutions

CALCULUS (CLOTH)

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