Domanda del libro di testo
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.7.51
Evaluate the integrals in Exercises 41–60.
51. ∫(from ln2 to ln4)coth(x)dx
Guida verificata passo dopo passo1
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).
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06:21
Properties of Functions
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.
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5:50Asymptotes of Hyperbolas
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.
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05:43Definition of the Definite Integral
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