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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.7.37b
Capitolo 7, Problema 7.7.37b

Verify the integration formulas in Exercises 37–40.
37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C

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Recall the definition of the hyperbolic secant function: \(\text{sech}(x) = \frac{1}{\cosh(x)}\) and the hyperbolic tangent function: \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\).
Set up the integral to verify: \(\int \text{sech}(x) \, dx\) and consider the substitution \(u = \tanh(x)\), since the derivative of \(\tanh(x)\) is related to \(\text{sech}^2(x)\).
Compute the derivative of \(u\): \(\frac{du}{dx} = \text{sech}^2(x)\), which suggests expressing \(\text{sech}(x)\) in terms of \(u\) and \(du\) might require manipulation.
Rewrite \(\text{sech}(x)\) as \(\frac{\text{sech}^2(x)}{\text{sech}(x)}\) and use the substitution \(u = \tanh(x)\) to express the integral in terms of \(u\) and \(du\).
Recognize that the integral transforms into \(\int \frac{1}{\sqrt{1 - u^2}} \, du\), which corresponds to the inverse sine function, leading to the result \(\sin^{-1}(\tanh(x)) + C\).

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Hyperbolic Functions

Hyperbolic functions like sech(x) and tanh(x) are analogs of trigonometric functions but based on exponential functions. Understanding their definitions and properties is essential for integrating expressions involving these functions.
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Asymptotes of Hyperbolas

Integration Techniques for Hyperbolic Functions

Integrating hyperbolic functions often involves substitution or recognizing derivatives of related functions. For example, knowing that the derivative of tanh(x) is sech²(x) helps in manipulating integrals involving sech(x).
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Asymptotes of Hyperbolas

Inverse Hyperbolic and Trigonometric Functions

The integral result involves sin⁻¹(tanh x), an inverse trigonometric function applied to a hyperbolic function. Understanding the relationship between inverse trigonometric and hyperbolic functions aids in verifying and interpreting such integrals.
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Derivatives of Other Inverse Trigonometric Functions
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