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

Verified step by step guidance

1. Recall the definition of the hyperbolic secant function: \(\text{sech}(x) = \frac{1}{\cosh(x)}\) and the hyperbolic tangent function: \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\).
2. 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 \(\text{sech}^2(x)\).
3. 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.
4. 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 \(du\).
5. 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 \(\sin^{-1}(\tanh(x)) + C\).

Key Concepts

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.

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).

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.