Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
69. ∫(from 5/4 to 2)dx/(1-x²)
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 7.7.37b
Verify the integration formulas in Exercises 37–40.
37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C
Verified step by step guidance1
Recall the definition of the hyperbolic secant function: \(\text{sech}(x) = \frac{1}{\cosh(x)}\) and the hyperbolic tangent function: \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\).
Set up the integral to verify: \(\int \text{sech}(x) \, dx\) and consider the substitution \(u = \tanh(x)\), since the derivative of \(\tanh(x)\) is related to \(\text{sech}^2(x)\).
Compute the derivative of \(u\): \(\frac{du}{dx} = \text{sech}^2(x)\), which suggests expressing \(\text{sech}(x)\) in terms of \(u\) and \(du\) might require manipulation.
Rewrite \(\text{sech}(x)\) as \(\frac{\text{sech}^2(x)}{\text{sech}(x)}\) and use the substitution \(u = \tanh(x)\) to express the integral in terms of \(u\) and \(du\).
Recognize that the integral transforms into \(\int \frac{1}{\sqrt{1 - u^2}} \, du\), which corresponds to the inverse sine function, leading to the result \(\sin^{-1}(\tanh(x)) + C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions
Hyperbolic functions like sech(x) and tanh(x) are analogs of trigonometric functions but based on exponential functions. Understanding their definitions and properties is essential for integrating expressions involving these functions.
Recommended video:
5:50
Asymptotes of Hyperbolas
Integration Techniques for Hyperbolic Functions
Integrating hyperbolic functions often involves substitution or recognizing derivatives of related functions. For example, knowing that the derivative of tanh(x) is sech²(x) helps in manipulating integrals involving sech(x).
Recommended video:
5:50Asymptotes of Hyperbolas
Inverse Hyperbolic and Trigonometric Functions
The integral result involves sin⁻¹(tanh x), an inverse trigonometric function applied to a hyperbolic function. Understanding the relationship between inverse trigonometric and hyperbolic functions aids in verifying and interpreting such integrals.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
Textbook Question
In Exercises 1–4, solve for t.
2. b. e^(kt) = 10
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
6. b. arccsc(-2/√3)
Textbook Question
3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)