Evaluate the integrals in Exercises 41–60.
47. ∫sech²(x - 1/2)dx

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Recognize that the integral involves the function \(\operatorname{sech}^2(x - \frac{1}{2})\). Recall that \(\operatorname{sech}(x) = \frac{1}{\cosh(x)}\) and that the derivative of \(\tanh(x)\) is \(\operatorname{sech}^2(x)\).

Use the substitution method by letting \(u = x - \frac{1}{2}\). Then, the differential \(du = dx\).

Rewrite the integral in terms of \(u\): \(\int \operatorname{sech}^2(u) \, du\).

Recall the antiderivative formula: \(\int \operatorname{sech}^2(u) \, du = \tanh(u) + C\), where \(C\) is the constant of integration.

Substitute back \(u = x - \frac{1}{2}\) to express the answer in terms of \(x\): \(\tanh\left(x - \frac{1}{2}\right) + C\).

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주요 개념

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Hyperbolic Functions

Hyperbolic functions, such as sech(x), sinh(x), and cosh(x), are analogs of trigonometric functions but based on hyperbolas. The function sech(x) is defined as 1/cosh(x), and understanding their properties and derivatives is essential for integrating expressions involving them.

Integration of Hyperbolic Function Squares

Integrating the square of hyperbolic functions like sech²(x) often involves recognizing standard integral forms. For example, the integral of sech²(u) du is tanh(u) + C, similar to how the integral of sec²(x) dx is tan(x) + C in trigonometry.

Substitution Method in Integration

When the integrand contains a function of (x - a), substitution simplifies the integral. Setting u = x - 1/2 transforms the integral into a standard form, allowing direct application of known integral formulas for hyperbolic functions.

Evaluate the integrals in Exercises 41–60.47. ∫sech²(x - 1/2)dx