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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.108b
Capitolo 7, Problema 7.3.108b

"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).


b. Use the identity for sinh(2u) to show that 2 / sinh(2u) = sech² u / tanh u."

Guida verificata passo dopo passo
1
Recall the double-angle identity for hyperbolic sine: \(\sinh(2u) = 2 \sinh u \cosh u\).
Start with the expression \(\frac{2}{\sinh(2u)}\) and substitute the identity: \(\frac{2}{2 \sinh u \cosh u} = \frac{1}{\sinh u \cosh u}\).
Express \(\frac{1}{\sinh u \cosh u}\) in terms of \(\tanh u\) and \(\operatorname{sech} u\) by rewriting the denominator: \(\sinh u = \frac{\tanh u}{\operatorname{sech} u}\) and \(\cosh u = \frac{1}{\operatorname{sech} u}\).
Use the definitions \(\tanh u = \frac{\sinh u}{\cosh u}\) and \(\operatorname{sech} u = \frac{1}{\cosh u}\) to rewrite the expression \(\frac{1}{\sinh u \cosh u}\) as \(\frac{\operatorname{sech}^2 u}{\tanh u}\).
Conclude that \(\frac{2}{\sinh(2u)} = \frac{\operatorname{sech}^2 u}{\tanh u}\), as required.

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Hyperbolic Functions and Their Identities

Hyperbolic functions like sinh, cosh, and tanh are analogs of trigonometric functions but for a hyperbola. Key identities, such as sinh(2u) = 2 sinh u cosh u, help simplify expressions and are essential for manipulating integrals involving hyperbolic functions.
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Verifying Trig Equations as Identities

Integration of Hyperbolic Functions

Integrating hyperbolic functions often involves substitution and using their identities to rewrite the integrand. For example, integrating csch x requires expressing it in terms of sinh x and applying logarithmic integration techniques to arrive at the formula ∫ csch x dx = ln |tanh(x/2)| + C.
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Asymptotes of Hyperbolas

Algebraic Manipulation of Hyperbolic Expressions

Proving identities like 2 / sinh(2u) = sech² u / tanh u requires careful algebraic manipulation using definitions: sech u = 1/cosh u and tanh u = sinh u / cosh u. Rearranging and substituting these expressions is crucial to verify the given identity.
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Asymptotes of Hyperbolas
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Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


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Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

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Projection sensitivity

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