Many formulas There are several ways to express the indefinite integral of sech x.


b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

Verified step by step guidance

1. Start with the integral you want to solve: \(\int \text{sech}\,x \, dx\).
2. Use the hint to rewrite \(\text{sech}\,x\) as \(\frac{\text{sech}^2 x}{\sqrt{1 - \tanh^2 x}}\). So the integral becomes \(\int \frac{\text{sech}^2 x}{\sqrt{1 - \tanh^2 x}} \, dx\).
3. Recognize that the derivative of \(\tanh x\) is \(\text{sech}^2 x\), which suggests the substitution \(u = \tanh x\). Then, \(du = \text{sech}^2 x \, dx\).
4. Rewrite the integral in terms of \(u\): it becomes \(\int \frac{1}{\sqrt{1 - u^2}} \, du\).
5. Recall that \(\int \frac{1}{\sqrt{1 - u^2}} \, du = \sin^{-1} u + C\). Substitute back \(u = \tanh x\) to get the final expression: \(\sin^{-1}(\tanh x) + C\).

Key Concepts

Hyperbolic Functions and Their Identities

Hyperbolic functions like sech x and tanh x are analogs of trigonometric functions but based on exponential definitions. Understanding identities such as 1 - tanh² x = sech² x is crucial for manipulating expressions and simplifying integrals involving these functions.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. By expressing the integral in terms of a new variable (e.g., u = tanh x), the integral becomes easier to evaluate, often transforming complicated expressions into standard forms.

Inverse Hyperbolic and Trigonometric Functions

Inverse functions like sin⁻¹ and tanh⁻¹ help express antiderivatives in closed form. Recognizing when an integral corresponds to an inverse trigonometric function, such as sin⁻¹(tanh x), allows for correct evaluation and interpretation of indefinite integrals.