(a) Explanation Proof: Consider the following expression, cosh − 1 x = ln ( x + x 2 − 1 ) , x ≥ 1 Suppose that, y = cosh − 1 x x = cosh y x = e y + e − y 2 2 x = e y + 1 e y Further simplification, e 2 y + 1 = 2 x e y e 2 y − 2 x e y + 1 = 0 ( e y ) 2 − 2 x e y + 1 = 0 Use quadratic formula to solve the equation. e y = − ( − 2 x ) ± ( − 2 x ) 2 − 4 ( 1 ) ( 1 ) 2 e y = 2 x ± 4 ( (b) To determine To prove: the equation tanh − 1 x = 1 2 ln 1 + x 1 − x , | x | < 1 .

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 7.8, Problem 59PS

(a)

To determine

To prove:the equation cosh−1x=ln(x+x2−1),x≥1 .

(b)

To determine

To prove: the equation tanh−1x=12ln1+x1−x,|x|<1 .

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Chapter 7.8, Problem 58PS

Chapter 7.8, Problem 60PS

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Calculus

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To prove: the equation cosh − 1 x = ln ( x + x 2 − 1 ) , x ≥ 1 .

Solution Summary: The author explains how to solve the equation using a quadratic formula.