Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
27. y=arccsc(x²+1)
Textbook Question
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
8. cosh(3x) - sinh(3x)
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 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
27. y = θ(sin(lnθ) + cos(lnθ))
1
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Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
23. y = (x²+1)sech(ln x)
(Hint: Before differentiating, express in terms of exponentials and simplify.)