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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Key Concepts

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

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.

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