Evaluate the integrals in Exercises 41–60.
45. ∫tanh(x/7)dx

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Recall that the integral of the hyperbolic tangent function \( \tanh(u) \) with respect to \( u \) is \( \int \tanh(u) \, du = \ln|\cosh(u)| + C \).

Identify the inner function inside the hyperbolic tangent: here, \( u = \frac{x}{7} \).

Use substitution: let \( u = \frac{x}{7} \), then \( du = \frac{1}{7} dx \) or equivalently \( dx = 7 \, du \).

Rewrite the integral in terms of \( u \): \( \int \tanh\left(\frac{x}{7}\right) dx = \int \tanh(u) \cdot 7 \, du = 7 \int \tanh(u) \, du \).

Integrate using the known formula: \( 7 \int \tanh(u) \, du = 7 \ln|\cosh(u)| + C \), then substitute back \( u = \frac{x}{7} \) to get the final expression.

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Key Concepts

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

Hyperbolic Tangent Function (tanh)

The hyperbolic tangent function, tanh(x), is defined as (e^x - e^(-x)) / (e^x + e^(-x)). It is an odd function with values ranging between -1 and 1. Understanding its properties helps in recognizing integrals involving tanh and applying appropriate integration techniques.

Integration of Hyperbolic Functions

Integrating hyperbolic functions often involves using their definitions or known antiderivatives. For tanh(x), the integral is ln(cosh(x)) + C. Recognizing these standard integrals simplifies the process and avoids complex substitutions.

Substitution Method in Integration

When the integrand contains a function of a linear expression like tanh(x/7), substitution helps simplify the integral. Setting u = x/7 transforms the integral into a standard form, making it easier to integrate and then revert to the original variable.

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Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = x² − 2x, x ≤ 1

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