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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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