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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 7.7.37a
Capitolo 7, Problema 7.7.37a

Verify the integration formulas in Exercises 37–40.
37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C

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Recall the definition of the hyperbolic secant function: \(\text{sech}(x) = \frac{1}{\cosh(x)}\) and the hyperbolic sine function: \(\sinh(x)\).
Set up the integral: \(\int \text{sech}(x) \, dx = \int \frac{1}{\cosh(x)} \, dx\).
Use the substitution method by letting \(u = \sinh(x)\), then compute the derivative \(\frac{du}{dx} = \cosh(x)\), which implies \(dx = \frac{du}{\cosh(x)}\).
Rewrite the integral in terms of \(u\): substituting \(dx\) and \(\text{sech}(x)\), the integral becomes \(\int \frac{1}{\cosh(x)} \cdot \frac{du}{\cosh(x)} = \int \frac{1}{\cosh^2(x)} du\).
Recognize that \(\frac{1}{\cosh^2(x)} = \text{sech}^2(x)\) and recall that \(\frac{d}{dx} \tanh(x) = \text{sech}^2(x)\), so the integral simplifies to \(\int \text{sech}(x) \, dx = \tan^{-1}(\sinh(x)) + C\) after back-substitution and using the inverse tangent relationship.

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Hyperbolic Functions

Hyperbolic functions like sinh(x) and sech(x) are analogs of trigonometric functions but based on exponential functions. Understanding their definitions and properties, such as sech(x) = 1/cosh(x), is essential for manipulating and integrating expressions involving them.
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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 tan⁻¹(sinh x) relates to sech(x) helps verify the integral formula by differentiation.
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Asymptotes of Hyperbolas

Inverse Trigonometric Functions and Their Derivatives

Inverse trigonometric functions like tan⁻¹(u) have specific derivative formulas, such as d/dx [tan⁻¹(u)] = u' / (1 + u²). Applying this to u = sinh x allows verification of the integral by differentiating the proposed antiderivative.
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Derivatives of Other Inverse Trigonometric Functions
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