Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
Chapter 7, Problem 7.7.39
Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C
Verified step by step guidance1
Identify the integral to verify: \(\int x \coth^{-1}(x) \, dx\) and the proposed formula: \(\frac{(x^2 - 1)}{2} \coth^{-1}(x) + \frac{x}{2} + C\).
Use integration by parts, where you let \(u = \coth^{-1}(x)\) and \(dv = x \, dx\). Then compute \(du\) and \(v\):
\[ u = \coth^{-1}(x) \implies du = \frac{-1}{1 - x^2} \, dx, \quad dv = x \, dx \implies v = \frac{x^2}{2} \]
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write:
\[ \int x \coth^{-1}(x) \, dx = \frac{x^2}{2} \coth^{-1}(x) - \int \frac{x^2}{2} \cdot \left( \frac{-1}{1 - x^2} \right) dx \]
Simplify the integral inside and solve it step-by-step, then combine all terms to verify that the result matches the given formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Hyperbolic Cotangent Function (coth⁻¹(x))
The inverse hyperbolic cotangent function, coth⁻¹(x), is the inverse of the hyperbolic cotangent. It is defined for |x| > 1 and can be expressed using logarithms. Understanding its domain and properties is essential for integrating expressions involving coth⁻¹(x).
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Derivatives of Other Inverse Trigonometric Functions
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals. This method is often used when integrating products like x * coth⁻¹(x), where one function is easily differentiable and the other integrable.
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Verification of Integration Formulas
Verifying an integration formula involves differentiating the proposed antiderivative to check if it yields the original integrand. This process confirms the correctness of the integral and constant of integration, ensuring the formula is valid for the given function.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx
1
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Textbook Question
82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is
L = ∫(from 1 to 2)√(1 + 1/x²)dx.
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
115. y = (x + 1)ˣ
Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
13. y = 6sinh(x/3)
Textbook Question
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
2. sinh x = 4/3