Evaluate the integrals in Exercises 41–60.
51. ∫(from ln2 to ln4)coth(x)dx

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Recall the definition of the hyperbolic cotangent function: \(\coth(x) = \frac{\cosh(x)}{\sinh(x)}\).

Recognize that the integral of \(\coth(x)\) can be expressed in terms of the natural logarithm of the hyperbolic sine function: \(\int \coth(x) \, dx = \ln|\sinh(x)| + C\).

Set up the definite integral using the antiderivative: \(\int_{\ln 2}^{\ln 4} \coth(x) \, dx = \left[ \ln|\sinh(x)| \right]_{\ln 2}^{\ln 4}\).

Evaluate the expression at the upper and lower limits: calculate \(\ln|\sinh(\ln 4)|\) and \(\ln|\sinh(\ln 2)|\) separately.

Subtract the lower limit evaluation from the upper limit evaluation to find the value of the definite integral: \(\ln|\sinh(\ln 4)| - \ln|\sinh(\ln 2)|\).

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Definition and Properties of the Hyperbolic Cotangent Function

The hyperbolic cotangent function, coth(x), is defined as cosh(x)/sinh(x). It is important to understand its behavior and domain, especially since it has singularities where sinh(x) = 0. Recognizing its derivative and integral forms helps in evaluating integrals involving coth(x).

Techniques for Integrating Hyperbolic Functions

Integrating hyperbolic functions often involves rewriting them in terms of exponential functions or using known integral formulas. For coth(x), the integral is ln|sinh(x)| + C, which simplifies the evaluation of definite integrals over intervals where sinh(x) is nonzero.

Evaluating Definite Integrals with Logarithmic Functions

When integrating functions like coth(x), the result often involves logarithmic expressions. Evaluating definite integrals requires substituting the limits into these logarithmic forms carefully, considering the domain and ensuring the arguments of the logarithms are positive to avoid undefined values.

Evaluate the integrals in Exercises 41–60.51. ∫(from ln2 to ln4)coth(x)dx