37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

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Recall the definitions and derivatives of hyperbolic functions: \( \frac{d}{dw} \tanh w = \text{sech}^2 w \) and \( \frac{d}{dw} \text{sech} w = -\text{sech} w \tanh w \).

Notice that the integrand is \( \text{sech}^2 w \tanh w \), which can be seen as \( \text{sech}^2 w \times \tanh w \). Consider if this expression resembles the derivative of a product or a function involving \( \tanh w \) or \( \text{sech} w \).

Try to express the integrand in terms of a derivative of a function. For example, check if it matches the derivative of \( -\text{sech} w \) because \( \frac{d}{dw}(-\text{sech} w) = \text{sech} w \tanh w \), which is close but not exactly the integrand.

Alternatively, use substitution: let \( u = \tanh w \), then \( du = \text{sech}^2 w \, dw \). Rewrite the integral in terms of \( u \) and \( du \).

After substitution, the integral becomes \( \int u \, du \), which is straightforward to integrate. Then, substitute back \( u = \tanh w \) to express the answer in terms of \( w \).

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주요 개념

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Hyperbolic Functions

Hyperbolic functions, such as sech (hyperbolic secant) and tanh (hyperbolic tangent), are analogs of trigonometric functions but based on hyperbolas. Understanding their definitions and properties is essential for manipulating and integrating expressions involving these functions.

Integration Techniques for Hyperbolic Functions

Integrating products of hyperbolic functions often involves recognizing derivatives of one function within the integrand or using substitution. Familiarity with derivatives like d/dx(tanh x) = sech² x helps simplify and evaluate such integrals efficiently.

Substitution Method

The substitution method involves changing variables to simplify an integral. By identifying a part of the integrand as a derivative of another function, substitution transforms the integral into a basic form that is easier to evaluate.

37–56. Integrals Evaluate each integral.∫ sech² w tanh w dw