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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