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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